Algebra Examples

$(80, 300)$ , $(110, 375)$ , $(130, 450)$

Step 1

Use the standard form of a quadratic equation $y = a x^{2} + b x + c$ as the starting point for finding the equation through the three points.

$y = a x^{2} + b x + c$

Step 2

Create a system of equations by substituting the $x$ and $y$ values of each point into the standard formula of a quadratic equation to create the three equation system.

$300 = a {(80)}^{2} + b (80) + c, 375 = a {(110)}^{2} + b (110) + c, 450 = a {(130)}^{2} + b (130) + c$

Step 3

Solve the system of equations to find the values of $a$ , $b$ , and $c$ .

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Step 3.1

Solve for $c$ in $300 = a \cdot 80^{2} + b (80) + c$ .

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Step 3.1.1

Rewrite the equation as $a \cdot 80^{2} + b (80) + c = 300$ .

$a \cdot 80^{2} + b (80) + c = 300$

$375 = a \cdot 110^{2} + b (110) + c$

$450 = a \cdot 130^{2} + b (130) + c$

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Raise $80$ to the power of $2$ .

$a \cdot 6400 + b (80) + c = 300$

$375 = a \cdot 110^{2} + b (110) + c$

$450 = a \cdot 130^{2} + b (130) + c$

Step 3.1.2.2

Move $6400$ to the left of $a$ .

$6400 \cdot a + b (80) + c = 300$

$375 = a \cdot 110^{2} + b (110) + c$

$450 = a \cdot 130^{2} + b (130) + c$

Step 3.1.2.3

Move $80$ to the left of $b$ .

$6400 a + 80 b + c = 300$

$375 = a \cdot 110^{2} + b (110) + c$

$450 = a \cdot 130^{2} + b (130) + c$

$6400 a + 80 b + c = 300$

$375 = a \cdot 110^{2} + b (110) + c$

$450 = a \cdot 130^{2} + b (130) + c$

Step 3.1.3

Move all terms not containing $c$ to the right side of the equation.

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Step 3.1.3.1

Subtract $6400 a$ from both sides of the equation.

$80 b + c = 300 - 6400 a$

$375 = a \cdot 110^{2} + b (110) + c$

$450 = a \cdot 130^{2} + b (130) + c$

Step 3.1.3.2

Subtract $80 b$ from both sides of the equation.

$c = 300 - 6400 a - 80 b$

$375 = a \cdot 110^{2} + b (110) + c$

$450 = a \cdot 130^{2} + b (130) + c$

$c = 300 - 6400 a - 80 b$

$375 = a \cdot 110^{2} + b (110) + c$

$450 = a \cdot 130^{2} + b (130) + c$

$c = 300 - 6400 a - 80 b$

$375 = a \cdot 110^{2} + b (110) + c$

$450 = a \cdot 130^{2} + b (130) + c$

Step 3.2

Replace all occurrences of $c$ with $300 - 6400 a - 80 b$ in each equation.

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Step 3.2.1

Replace all occurrences of $c$ in $375 = a \cdot 110^{2} + b (110) + c$ with $300 - 6400 a - 80 b$ .

$375 = a \cdot 110^{2} + b (110) + 300 - 6400 a - 80 b$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + c$

Step 3.2.2

Simplify $375 = a \cdot 110^{2} + b (110) + 300 - 6400 a - 80 b$ .

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Remove parentheses.

$375 = a \cdot 110^{2} + b (110) + 300 - 6400 a - 80 b$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + c$

$375 = a \cdot 110^{2} + b (110) + 300 - 6400 a - 80 b$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + c$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Simplify $a \cdot 110^{2} + b (110) + 300 - 6400 a - 80 b$ .

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Step 3.2.2.2.1.1

Simplify each term.

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Step 3.2.2.2.1.1.1

Raise $110$ to the power of $2$ .

$375 = a \cdot 12100 + b (110) + 300 - 6400 a - 80 b$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + c$

Step 3.2.2.2.1.1.2

Move $12100$ to the left of $a$ .

$375 = 12100 \cdot a + b (110) + 300 - 6400 a - 80 b$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + c$

Step 3.2.2.2.1.1.3

Move $110$ to the left of $b$ .

$375 = 12100 a + 110 b + 300 - 6400 a - 80 b$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + c$

$375 = 12100 a + 110 b + 300 - 6400 a - 80 b$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + c$

Step 3.2.2.2.1.2

Simplify by adding terms.

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Step 3.2.2.2.1.2.1

Subtract $6400 a$ from $12100 a$ .

$375 = 5700 a + 110 b + 300 - 80 b$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + c$

Step 3.2.2.2.1.2.2

Subtract $80 b$ from $110 b$ .

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + c$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + c$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + c$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + c$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + c$

Step 3.2.3

Replace all occurrences of $c$ in $450 = a \cdot 130^{2} + b (130) + c$ with $300 - 6400 a - 80 b$ .

$450 = a \cdot 130^{2} + b (130) + 300 - 6400 a - 80 b$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.2.4

Simplify $450 = a \cdot 130^{2} + b (130) + 300 - 6400 a - 80 b$ .

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Step 3.2.4.1

Simplify the left side.

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Step 3.2.4.1.1

Remove parentheses.

$450 = a \cdot 130^{2} + b (130) + 300 - 6400 a - 80 b$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$450 = a \cdot 130^{2} + b (130) + 300 - 6400 a - 80 b$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.2.4.2

Simplify the right side.

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Step 3.2.4.2.1

Simplify $a \cdot 130^{2} + b (130) + 300 - 6400 a - 80 b$ .

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Step 3.2.4.2.1.1

Simplify each term.

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Step 3.2.4.2.1.1.1

Raise $130$ to the power of $2$ .

$450 = a \cdot 16900 + b (130) + 300 - 6400 a - 80 b$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.2.4.2.1.1.2

Move $16900$ to the left of $a$ .

$450 = 16900 \cdot a + b (130) + 300 - 6400 a - 80 b$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.2.4.2.1.1.3

Move $130$ to the left of $b$ .

$450 = 16900 a + 130 b + 300 - 6400 a - 80 b$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$450 = 16900 a + 130 b + 300 - 6400 a - 80 b$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.2.4.2.1.2

Simplify by adding terms.

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Step 3.2.4.2.1.2.1

Subtract $6400 a$ from $16900 a$ .

$450 = 10500 a + 130 b + 300 - 80 b$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.2.4.2.1.2.2

Subtract $80 b$ from $130 b$ .

$450 = 10500 a + 50 b + 300$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$450 = 10500 a + 50 b + 300$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$450 = 10500 a + 50 b + 300$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$450 = 10500 a + 50 b + 300$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$450 = 10500 a + 50 b + 300$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$450 = 10500 a + 50 b + 300$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3

Solve for $a$ in $450 = 10500 a + 50 b + 300$ .

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Step 3.3.1

Rewrite the equation as $10500 a + 50 b + 300 = 450$ .

$10500 a + 50 b + 300 = 450$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.2

Move all terms not containing $a$ to the right side of the equation.

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Step 3.3.2.1

Subtract $50 b$ from both sides of the equation.

$10500 a + 300 = 450 - 50 b$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.2.2

Subtract $300$ from both sides of the equation.

$10500 a = 450 - 50 b - 300$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.2.3

Subtract $300$ from $450$ .

$10500 a = - 50 b + 150$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$10500 a = - 50 b + 150$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.3

Divide each term in $10500 a = - 50 b + 150$ by $10500$ and simplify.

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Step 3.3.3.1

Divide each term in $10500 a = - 50 b + 150$ by $10500$ .

$\frac{10500 a}{10500} = \frac{- 50 b}{10500} + \frac{150}{10500}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $10500$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{10500 a}{10500} = \frac{- 50 b}{10500} + \frac{150}{10500}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 50 b}{10500} + \frac{150}{10500}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$a = \frac{- 50 b}{10500} + \frac{150}{10500}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$a = \frac{- 50 b}{10500} + \frac{150}{10500}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Simplify each term.

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Step 3.3.3.3.1.1

Cancel the common factor of $- 50$ and $10500$ .

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Step 3.3.3.3.1.1.1

Factor $50$ out of $- 50 b$ .

$a = \frac{50 (- b)}{10500} + \frac{150}{10500}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.3.3.1.1.2

Cancel the common factors.

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Step 3.3.3.3.1.1.2.1

Factor $50$ out of $10500$ .

$a = \frac{50 (- b)}{50 (210)} + \frac{150}{10500}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.3.3.1.1.2.2

Cancel the common factor.

$a = \frac{50 (- b)}{50 \cdot 210} + \frac{150}{10500}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.3.3.1.1.2.3

Rewrite the expression.

$a = \frac{- b}{210} + \frac{150}{10500}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$a = \frac{- b}{210} + \frac{150}{10500}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$a = \frac{- b}{210} + \frac{150}{10500}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.3.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b}{210} + \frac{150}{10500}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.3.3.1.3

Cancel the common factor of $150$ and $10500$ .

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Step 3.3.3.3.1.3.1

Factor $150$ out of $150$ .

$a = - \frac{b}{210} + \frac{150 (1)}{10500}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.3.3.1.3.2

Cancel the common factors.

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Step 3.3.3.3.1.3.2.1

Factor $150$ out of $10500$ .

$a = - \frac{b}{210} + \frac{150 \cdot 1}{150 \cdot 70}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.3.3.1.3.2.2

Cancel the common factor.

$a = - \frac{b}{210} + \frac{150 \cdot 1}{150 \cdot 70}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.3.3.3.1.3.2.3

Rewrite the expression.

$a = - \frac{b}{210} + \frac{1}{70}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$a = - \frac{b}{210} + \frac{1}{70}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$a = - \frac{b}{210} + \frac{1}{70}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$a = - \frac{b}{210} + \frac{1}{70}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$a = - \frac{b}{210} + \frac{1}{70}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$a = - \frac{b}{210} + \frac{1}{70}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

$a = - \frac{b}{210} + \frac{1}{70}$

$375 = 5700 a + 30 b + 300$

$c = 300 - 6400 a - 80 b$

Step 3.4

Replace all occurrences of $a$ with $- \frac{b}{210} + \frac{1}{70}$ in each equation.

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Step 3.4.1

Replace all occurrences of $a$ in $375 = 5700 a + 30 b + 300$ with $- \frac{b}{210} + \frac{1}{70}$ .

$375 = 5700 (- \frac{b}{210} + \frac{1}{70}) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $5700 (- \frac{b}{210} + \frac{1}{70}) + 30 b + 300$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Apply the distributive property.

$375 = 5700 (- \frac{b}{210}) + 5700 (\frac{1}{70}) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.1.2

Cancel the common factor of $30$ .

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Step 3.4.2.1.1.2.1

Move the leading negative in $- \frac{b}{210}$ into the numerator.

$375 = 5700 (\frac{- b}{210}) + 5700 (\frac{1}{70}) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.1.2.2

Factor $30$ out of $5700$ .

$375 = 30 (190) (\frac{- b}{210}) + 5700 (\frac{1}{70}) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.1.2.3

Factor $30$ out of $210$ .

$375 = 30 \cdot (190 (\frac{- b}{30 \cdot 7})) + 5700 (\frac{1}{70}) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.1.2.4

Cancel the common factor.

$375 = 30 \cdot (190 (\frac{- b}{30 \cdot 7})) + 5700 (\frac{1}{70}) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.1.2.5

Rewrite the expression.

$375 = 190 (\frac{- b}{7}) + 5700 (\frac{1}{70}) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

$375 = 190 (\frac{- b}{7}) + 5700 (\frac{1}{70}) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.1.3

Combine $190$ and $\frac{- b}{7}$ .

$375 = \frac{190 (- b)}{7} + 5700 (\frac{1}{70}) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.1.4

Multiply $- 1$ by $190$ .

$375 = \frac{- 190 b}{7} + 5700 (\frac{1}{70}) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.1.5

Cancel the common factor of $10$ .

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Step 3.4.2.1.1.5.1

Factor $10$ out of $5700$ .

$375 = \frac{- 190 b}{7} + 10 (570) (\frac{1}{70}) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.1.5.2

Factor $10$ out of $70$ .

$375 = \frac{- 190 b}{7} + 10 \cdot (570 (\frac{1}{10 \cdot 7})) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.1.5.3

Cancel the common factor.

$375 = \frac{- 190 b}{7} + 10 \cdot (570 (\frac{1}{10 \cdot 7})) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.1.5.4

Rewrite the expression.

$375 = \frac{- 190 b}{7} + 570 (\frac{1}{7}) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

$375 = \frac{- 190 b}{7} + 570 (\frac{1}{7}) + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.1.6

Combine $570$ and $\frac{1}{7}$ .

$375 = \frac{- 190 b}{7} + \frac{570}{7} + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.1.7

Move the negative in front of the fraction.

$375 = - \frac{190 b}{7} + \frac{570}{7} + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

$375 = - \frac{190 b}{7} + \frac{570}{7} + 30 b + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.2

To write $30 b$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$375 = - \frac{190 b}{7} + 30 b \cdot \frac{7}{7} + \frac{570}{7} + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.3

Combine $30 b$ and $\frac{7}{7}$ .

$375 = - \frac{190 b}{7} + \frac{30 b \cdot 7}{7} + \frac{570}{7} + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.4

Combine the numerators over the common denominator.

$375 = \frac{- 190 b + 30 b \cdot 7}{7} + \frac{570}{7} + 300$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.5

Combine the numerators over the common denominator.

$375 = 300 + \frac{- 190 b + 30 b \cdot 7 + 570}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.6

Multiply $7$ by $30$ .

$375 = 300 + \frac{- 190 b + 210 b + 570}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.7

Add $- 190 b$ and $210 b$ .

$375 = 300 + \frac{20 b + 570}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.8

Factor $10$ out of $20 b + 570$ .

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Step 3.4.2.1.8.1

Factor $10$ out of $20 b$ .

$375 = 300 + \frac{10 (2 b) + 570}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.8.2

Factor $10$ out of $570$ .

$375 = 300 + \frac{10 (2 b) + 10 (57)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.8.3

Factor $10$ out of $10 (2 b) + 10 (57)$ .

$375 = 300 + \frac{10 (2 b + 57)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

$375 = 300 + \frac{10 (2 b + 57)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.9

To write $300$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$375 = 300 \cdot \frac{7}{7} + \frac{10 (2 b + 57)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.10

Simplify terms.

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Step 3.4.2.1.10.1

Combine $300$ and $\frac{7}{7}$ .

$375 = \frac{300 \cdot 7}{7} + \frac{10 (2 b + 57)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.10.2

Combine the numerators over the common denominator.

$375 = \frac{300 \cdot 7 + 10 (2 b + 57)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

$375 = \frac{300 \cdot 7 + 10 (2 b + 57)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.11

Simplify the numerator.

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Step 3.4.2.1.11.1

Factor $10$ out of $300 \cdot 7 + 10 (2 b + 57)$ .

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Step 3.4.2.1.11.1.1

Factor $10$ out of $300 \cdot 7$ .

$375 = \frac{10 (30 \cdot 7) + 10 (2 b + 57)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.11.1.2

Factor $10$ out of $10 (30 \cdot 7) + 10 (2 b + 57)$ .

$375 = \frac{10 (30 \cdot 7 + 2 b + 57)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

$375 = \frac{10 (30 \cdot 7 + 2 b + 57)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.11.2

Multiply $30$ by $7$ .

$375 = \frac{10 (210 + 2 b + 57)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.2.1.11.3

Add $210$ and $57$ .

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 6400 a - 80 b$

Step 3.4.3

Replace all occurrences of $a$ in $c = 300 - 6400 a - 80 b$ with $- \frac{b}{210} + \frac{1}{70}$ .

$c = 300 - 6400 (- \frac{b}{210} + \frac{1}{70}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Simplify $300 - 6400 (- \frac{b}{210} + \frac{1}{70}) - 80 b$ .

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Step 3.4.4.1.1

Simplify each term.

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Step 3.4.4.1.1.1

Apply the distributive property.

$c = 300 - 6400 (- \frac{b}{210}) - 6400 (\frac{1}{70}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.1.2

Cancel the common factor of $10$ .

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Step 3.4.4.1.1.2.1

Move the leading negative in $- \frac{b}{210}$ into the numerator.

$c = 300 - 6400 \frac{- b}{210} - 6400 (\frac{1}{70}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.1.2.2

Factor $10$ out of $- 6400$ .

$c = 300 + 10 (- 640) (\frac{- b}{210}) - 6400 (\frac{1}{70}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.1.2.3

Factor $10$ out of $210$ .

$c = 300 + 10 \cdot (- 640 \frac{- b}{10 \cdot 21}) - 6400 (\frac{1}{70}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.1.2.4

Cancel the common factor.

$c = 300 + 10 \cdot (- 640 \frac{- b}{10 \cdot 21}) - 6400 (\frac{1}{70}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.1.2.5

Rewrite the expression.

$c = 300 - 640 \frac{- b}{21} - 6400 (\frac{1}{70}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 - 640 \frac{- b}{21} - 6400 (\frac{1}{70}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.1.3

Combine $- 640$ and $\frac{- b}{21}$ .

$c = 300 + \frac{- 640 (- b)}{21} - 6400 (\frac{1}{70}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.1.4

Multiply $- 1$ by $- 640$ .

$c = 300 + \frac{640 b}{21} - 6400 (\frac{1}{70}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.1.5

Cancel the common factor of $10$ .

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Step 3.4.4.1.1.5.1

Factor $10$ out of $- 6400$ .

$c = 300 + \frac{640 b}{21} + 10 (- 640) (\frac{1}{70}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.1.5.2

Factor $10$ out of $70$ .

$c = 300 + \frac{640 b}{21} + 10 \cdot (- 640 \frac{1}{10 \cdot 7}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.1.5.3

Cancel the common factor.

$c = 300 + \frac{640 b}{21} + 10 \cdot (- 640 \frac{1}{10 \cdot 7}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.1.5.4

Rewrite the expression.

$c = 300 + \frac{640 b}{21} - 640 (\frac{1}{7}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 + \frac{640 b}{21} - 640 (\frac{1}{7}) - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.1.6

Combine $- 640$ and $\frac{1}{7}$ .

$c = 300 + \frac{640 b}{21} + \frac{- 640}{7} - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.1.7

Move the negative in front of the fraction.

$c = 300 + \frac{640 b}{21} - \frac{640}{7} - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 300 + \frac{640 b}{21} - \frac{640}{7} - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.2

To write $300$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$c = \frac{640 b}{21} + 300 \cdot \frac{7}{7} - \frac{640}{7} - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.3

Combine $300$ and $\frac{7}{7}$ .

$c = \frac{640 b}{21} + \frac{300 \cdot 7}{7} - \frac{640}{7} - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.4

Combine the numerators over the common denominator.

$c = \frac{640 b}{21} + \frac{300 \cdot 7 - 640}{7} - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.5

Simplify the numerator.

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Step 3.4.4.1.5.1

Multiply $300$ by $7$ .

$c = \frac{640 b}{21} + \frac{2100 - 640}{7} - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.5.2

Subtract $640$ from $2100$ .

$c = \frac{640 b}{21} + \frac{1460}{7} - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = \frac{640 b}{21} + \frac{1460}{7} - 80 b$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.6

To write $- 80 b$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

$c = \frac{640 b}{21} - 80 b \cdot \frac{21}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.7

Simplify terms.

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Step 3.4.4.1.7.1

Combine $- 80 b$ and $\frac{21}{21}$ .

$c = \frac{640 b}{21} + \frac{- 80 b \cdot 21}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.7.2

Combine the numerators over the common denominator.

$c = \frac{640 b - 80 b \cdot 21}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = \frac{640 b - 80 b \cdot 21}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.8

Simplify each term.

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Step 3.4.4.1.8.1

Simplify the numerator.

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Step 3.4.4.1.8.1.1

Factor $80 b$ out of $640 b - 80 b \cdot 21$ .

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Step 3.4.4.1.8.1.1.1

Factor $80 b$ out of $640 b$ .

$c = \frac{80 b (8) - 80 b \cdot 21}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.8.1.1.2

Factor $80 b$ out of $- 80 b \cdot 21$ .

$c = \frac{80 b (8) + 80 b (- 1 \cdot 21)}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.8.1.1.3

Factor $80 b$ out of $80 b (8) + 80 b (- 1 \cdot 21)$ .

$c = \frac{80 b (8 - 1 \cdot 21)}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = \frac{80 b (8 - 1 \cdot 21)}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.8.1.2

Multiply $- 1$ by $21$ .

$c = \frac{80 b (8 - 21)}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.8.1.3

Subtract $21$ from $8$ .

$c = \frac{80 b \cdot - 13}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.8.1.4

Multiply $- 13$ by $80$ .

$c = \frac{- 1040 b}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = \frac{- 1040 b}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.4.4.1.8.2

Move the negative in front of the fraction.

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$375 = \frac{10 (2 b + 267)}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5

Solve for $b$ in $375 = \frac{10 (2 b + 267)}{7}$ .

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Step 3.5.1

Rewrite the equation as $\frac{10 (2 b + 267)}{7} = 375$ .

$\frac{10 (2 b + 267)}{7} = 375$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.2

Multiply both sides by $7$ .

$\frac{10 (2 b + 267)}{7} \cdot 7 = 375 \cdot 7$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.3

Simplify.

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Step 3.5.3.1

Simplify the left side.

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Step 3.5.3.1.1

Simplify $\frac{10 (2 b + 267)}{7} \cdot 7$ .

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Step 3.5.3.1.1.1

Cancel the common factor of $7$ .

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Step 3.5.3.1.1.1.1

Cancel the common factor.

$\frac{10 (2 b + 267)}{7} \cdot 7 = 375 \cdot 7$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.3.1.1.1.2

Rewrite the expression.

$10 (2 b + 267) = 375 \cdot 7$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$10 (2 b + 267) = 375 \cdot 7$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.3.1.1.2

Apply the distributive property.

$10 (2 b) + 10 \cdot 267 = 375 \cdot 7$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.3.1.1.3

Multiply.

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Step 3.5.3.1.1.3.1

Multiply $2$ by $10$ .

$20 b + 10 \cdot 267 = 375 \cdot 7$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.3.1.1.3.2

Multiply $10$ by $267$ .

$20 b + 2670 = 375 \cdot 7$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$20 b + 2670 = 375 \cdot 7$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$20 b + 2670 = 375 \cdot 7$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$20 b + 2670 = 375 \cdot 7$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.3.2

Simplify the right side.

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Step 3.5.3.2.1

Multiply $375$ by $7$ .

$20 b + 2670 = 2625$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$20 b + 2670 = 2625$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$20 b + 2670 = 2625$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.4

Solve for $b$ .

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Step 3.5.4.1

Move all terms not containing $b$ to the right side of the equation.

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Step 3.5.4.1.1

Subtract $2670$ from both sides of the equation.

$20 b = 2625 - 2670$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.4.1.2

Subtract $2670$ from $2625$ .

$20 b = - 45$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$20 b = - 45$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.4.2

Divide each term in $20 b = - 45$ by $20$ and simplify.

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Step 3.5.4.2.1

Divide each term in $20 b = - 45$ by $20$ .

$\frac{20 b}{20} = \frac{- 45}{20}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.4.2.2

Simplify the left side.

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Step 3.5.4.2.2.1

Cancel the common factor of $20$ .

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Step 3.5.4.2.2.1.1

Cancel the common factor.

$\frac{20 b}{20} = \frac{- 45}{20}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 45}{20}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$b = \frac{- 45}{20}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$b = \frac{- 45}{20}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.4.2.3

Simplify the right side.

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Step 3.5.4.2.3.1

Cancel the common factor of $- 45$ and $20$ .

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Step 3.5.4.2.3.1.1

Factor $5$ out of $- 45$ .

$b = \frac{5 (- 9)}{20}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.4.2.3.1.2

Cancel the common factors.

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Step 3.5.4.2.3.1.2.1

Factor $5$ out of $20$ .

$b = \frac{5 \cdot - 9}{5 \cdot 4}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.4.2.3.1.2.2

Cancel the common factor.

$b = \frac{5 \cdot - 9}{5 \cdot 4}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.4.2.3.1.2.3

Rewrite the expression.

$b = \frac{- 9}{4}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$b = \frac{- 9}{4}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$b = \frac{- 9}{4}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.5.4.2.3.2

Move the negative in front of the fraction.

$b = - \frac{9}{4}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$b = - \frac{9}{4}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$b = - \frac{9}{4}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$b = - \frac{9}{4}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

$b = - \frac{9}{4}$

$c = - \frac{1040 b}{21} + \frac{1460}{7}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6

Replace all occurrences of $b$ with $- \frac{9}{4}$ in each equation.

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Step 3.6.1

Replace all occurrences of $b$ in $c = - \frac{1040 b}{21} + \frac{1460}{7}$ with $- \frac{9}{4}$ .

$c = - \frac{1040 (- \frac{9}{4})}{21} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $- \frac{1040 (- \frac{9}{4})}{21} + \frac{1460}{7}$ .

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Step 3.6.2.1.1

Simplify each term.

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Step 3.6.2.1.1.1

Simplify the numerator.

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Step 3.6.2.1.1.1.1

Multiply $- 1$ by $1040$ .

$c = - \frac{- 1040 (\frac{9}{4})}{21} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2.1.1.1.2

Combine $- 1040$ and $\frac{9}{4}$ .

$c = - \frac{\frac{- 1040 \cdot 9}{4}}{21} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = - \frac{\frac{- 1040 \cdot 9}{4}}{21} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2.1.1.2

Multiply $- 1040$ by $9$ .

$c = - \frac{\frac{- 9360}{4}}{21} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2.1.1.3

Divide $- 9360$ by $4$ .

$c = - \frac{- 2340}{21} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2.1.1.4

Cancel the common factor of $- 2340$ and $21$ .

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Step 3.6.2.1.1.4.1

Factor $3$ out of $- 2340$ .

$c = - \frac{3 (- 780)}{21} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2.1.1.4.2

Cancel the common factors.

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Step 3.6.2.1.1.4.2.1

Factor $3$ out of $21$ .

$c = - \frac{3 \cdot - 780}{3 \cdot 7} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2.1.1.4.2.2

Cancel the common factor.

$c = - \frac{3 \cdot - 780}{3 \cdot 7} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2.1.1.4.2.3

Rewrite the expression.

$c = - \frac{- 780}{7} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = - \frac{- 780}{7} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = - \frac{- 780}{7} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2.1.1.5

Move the negative in front of the fraction.

$c = \frac{780}{7} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2.1.1.6

Multiply $- - \frac{780}{7}$ .

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Step 3.6.2.1.1.6.1

Multiply $- 1$ by $- 1$ .

$c = 1 (\frac{780}{7}) + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2.1.1.6.2

Multiply $\frac{780}{7}$ by $1$ .

$c = \frac{780}{7} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = \frac{780}{7} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = \frac{780}{7} + \frac{1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2.1.2

Combine fractions.

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Step 3.6.2.1.2.1

Combine the numerators over the common denominator.

$c = \frac{780 + 1460}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2.1.2.2

Simplify the expression.

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Step 3.6.2.1.2.2.1

Add $780$ and $1460$ .

$c = \frac{2240}{7}$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.2.1.2.2.2

Divide $2240$ by $7$ .

$c = 320$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

$a = - \frac{b}{210} + \frac{1}{70}$

Step 3.6.3

Replace all occurrences of $b$ in $a = - \frac{b}{210} + \frac{1}{70}$ with $- \frac{9}{4}$ .

$a = - \frac{- \frac{9}{4}}{210} + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify $- \frac{- \frac{9}{4}}{210} + \frac{1}{70}$ .

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Step 3.6.4.1.1

Simplify each term.

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Step 3.6.4.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$a = - (- \frac{9}{4} \cdot \frac{1}{210}) + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.1.2

Cancel the common factor of $3$ .

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Step 3.6.4.1.1.2.1

Move the leading negative in $- \frac{9}{4}$ into the numerator.

$a = - (\frac{- 9}{4} \cdot \frac{1}{210}) + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.1.2.2

Factor $3$ out of $- 9$ .

$a = - (\frac{3 (- 3)}{4} \cdot \frac{1}{210}) + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.1.2.3

Factor $3$ out of $210$ .

$a = - (\frac{3 \cdot - 3}{4} \cdot \frac{1}{3 \cdot 70}) + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.1.2.4

Cancel the common factor.

$a = - (\frac{3 \cdot - 3}{4} \cdot \frac{1}{3 \cdot 70}) + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.1.2.5

Rewrite the expression.

$a = - (\frac{- 3}{4} \cdot \frac{1}{70}) + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

$a = - (\frac{- 3}{4} \cdot \frac{1}{70}) + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.1.3

Multiply $\frac{- 3}{4}$ by $\frac{1}{70}$ .

$a = - \frac{- 3}{4 \cdot 70} + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.1.4

Multiply $4$ by $70$ .

$a = - \frac{- 3}{280} + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.1.5

Move the negative in front of the fraction.

$a = \frac{3}{280} + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.1.6

Multiply $- - \frac{3}{280}$ .

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Step 3.6.4.1.1.6.1

Multiply $- 1$ by $- 1$ .

$a = 1 (\frac{3}{280}) + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.1.6.2

Multiply $\frac{3}{280}$ by $1$ .

$a = \frac{3}{280} + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

$a = \frac{3}{280} + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

$a = \frac{3}{280} + \frac{1}{70}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.2

To write $\frac{1}{70}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$a = \frac{3}{280} + \frac{1}{70} \cdot \frac{4}{4}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.3

Write each expression with a common denominator of $280$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.6.4.1.3.1

Multiply $\frac{1}{70}$ by $\frac{4}{4}$ .

$a = \frac{3}{280} + \frac{4}{70 \cdot 4}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.3.2

Multiply $70$ by $4$ .

$a = \frac{3}{280} + \frac{4}{280}$

$c = 320$

$b = - \frac{9}{4}$

$a = \frac{3}{280} + \frac{4}{280}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.4

Simplify the expression.

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Step 3.6.4.1.4.1

Combine the numerators over the common denominator.

$a = \frac{3 + 4}{280}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.4.2

Add $3$ and $4$ .

$a = \frac{7}{280}$

$c = 320$

$b = - \frac{9}{4}$

$a = \frac{7}{280}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.5

Cancel the common factor of $7$ and $280$ .

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Step 3.6.4.1.5.1

Factor $7$ out of $7$ .

$a = \frac{7 (1)}{280}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.5.2

Cancel the common factors.

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Step 3.6.4.1.5.2.1

Factor $7$ out of $280$ .

$a = \frac{7 \cdot 1}{7 \cdot 40}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.5.2.2

Cancel the common factor.

$a = \frac{7 \cdot 1}{7 \cdot 40}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.6.4.1.5.2.3

Rewrite the expression.

$a = \frac{1}{40}$

$c = 320$

$b = - \frac{9}{4}$

$a = \frac{1}{40}$

$c = 320$

$b = - \frac{9}{4}$

$a = \frac{1}{40}$

$c = 320$

$b = - \frac{9}{4}$

$a = \frac{1}{40}$

$c = 320$

$b = - \frac{9}{4}$

$a = \frac{1}{40}$

$c = 320$

$b = - \frac{9}{4}$

$a = \frac{1}{40}$

$c = 320$

$b = - \frac{9}{4}$

Step 3.7

List all of the solutions.

$a = \frac{1}{40}, c = 320, b = - \frac{9}{4}$

Step 4

Substitute the actual values of $a$ , $b$ , and $c$ into the formula for a quadratic equation to find the resulting equation.

$y = \frac{x^{2}}{40} - \frac{9 x}{4} + 320$

Step 5