Precalculus Examples

$4 x^{2} + y = 3$ , $- x - y = 11$ , $(\frac{- 7}{4}, \frac{- 37}{4})$

Step 1

Simplify $4 {(\frac{- 7}{4})}^{2} + \frac{- 37}{4}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Move the negative in front of the fraction.

$4 {(- \frac{7}{4})}^{2} + \frac{- 37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 1.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.2.1

Apply the product rule to $- \frac{7}{4}$ .

$4 ({(- 1)}^{2} {(\frac{7}{4})}^{2}) + \frac{- 37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 1.1.2.2

Apply the product rule to $\frac{7}{4}$ .

$4 ({(- 1)}^{2} (\frac{7^{2}}{4^{2}})) + \frac{- 37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

$4 ({(- 1)}^{2} (\frac{7^{2}}{4^{2}})) + \frac{- 37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 1.1.3

Raise $- 1$ to the power of $2$ .

$4 (1 (\frac{7^{2}}{4^{2}})) + \frac{- 37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 1.1.4

Multiply $\frac{7^{2}}{4^{2}}$ by $1$ .

$4 (\frac{7^{2}}{4^{2}}) + \frac{- 37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 1.1.5

Raise $7$ to the power of $2$ .

$4 (\frac{49}{4^{2}}) + \frac{- 37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 1.1.6

Raise $4$ to the power of $2$ .

$4 (\frac{49}{16}) + \frac{- 37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 1.1.7

Cancel the common factor of $4$ .

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Step 1.1.7.1

Factor $4$ out of $16$ .

$4 (\frac{49}{4 (4)}) + \frac{- 37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 1.1.7.2

Cancel the common factor.

$4 (\frac{49}{4 \cdot 4}) + \frac{- 37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 1.1.7.3

Rewrite the expression.

$\frac{49}{4} + \frac{- 37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

$\frac{49}{4} + \frac{- 37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 1.1.8

Move the negative in front of the fraction.

$\frac{49}{4} - \frac{37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

$\frac{49}{4} - \frac{37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 1.2

Combine fractions.

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Step 1.2.1

Combine the numerators over the common denominator.

$\frac{49 - 37}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Subtract $37$ from $49$ .

$\frac{12}{4} = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 1.2.2.2

Divide $12$ by $4$ .

$3 = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

$3 = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

$3 = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

$3 = 3$

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 2

Since $3 = 3$ , the equation will always be true.

Always true

$- (\frac{- 7}{4}) - (\frac{- 37}{4}) = 11$

Step 3

Simplify $- \frac{- 7}{4} - \frac{- 37}{4}$ .

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Step 3.1

Combine the numerators over the common denominator.

$\frac{7 + 37}{4} = 11$

Always true

Step 3.2

Simplify the expression.

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Step 3.2.1

Add $7$ and $37$ .

$\frac{44}{4} = 11$

Always true

Step 3.2.2

Divide $44$ by $4$ .

$11 = 11$

Always true

$11 = 11$

Always true

$11 = 11$

Always true

Step 4

Since $11 = 11$ , the equation will always be true.

Always true

Step 5

The ordered pair is a solution to the system of equations.

$(\frac{- 7}{4}, \frac{- 37}{4})$ is a solution