Algebra Examples

$x^{2} + y^{2} = 25$ $y = \frac{3}{4} x - \frac{25}{4}$

Step 1

Combine $\frac{3}{4}$ and $x$ .

$y = \frac{3 x}{4} - \frac{25}{4}$

$x^{2} + y^{2} = 25$

Step 2

Replace all occurrences of $y$ with $\frac{3 x}{4} - \frac{25}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x^{2} + y^{2} = 25$ with $\frac{3 x}{4} - \frac{25}{4}$ .

$x^{2} + {(\frac{3 x}{4} - \frac{25}{4})}^{2} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x^{2} + {(\frac{3 x}{4} - \frac{25}{4})}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(\frac{3 x}{4} - \frac{25}{4})}^{2}$ as $(\frac{3 x}{4} - \frac{25}{4}) (\frac{3 x}{4} - \frac{25}{4})$ .

$x^{2} + (\frac{3 x}{4} - \frac{25}{4}) (\frac{3 x}{4} - \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.2

Expand $(\frac{3 x}{4} - \frac{25}{4}) (\frac{3 x}{4} - \frac{25}{4})$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$x^{2} + \frac{3 x}{4} \cdot (\frac{3 x}{4} - \frac{25}{4}) - \frac{25}{4} \cdot (\frac{3 x}{4} - \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.2.2

Apply the distributive property.

$x^{2} + \frac{3 x}{4} \cdot \frac{3 x}{4} + \frac{3 x}{4} \cdot (- \frac{25}{4}) - \frac{25}{4} \cdot (\frac{3 x}{4} - \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.2.3

Apply the distributive property.

$x^{2} + \frac{3 x}{4} \cdot \frac{3 x}{4} + \frac{3 x}{4} \cdot (- \frac{25}{4}) - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$x^{2} + \frac{3 x}{4} \cdot \frac{3 x}{4} + \frac{3 x}{4} \cdot (- \frac{25}{4}) - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3

Simplify and combine like terms.

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Step 2.2.1.1.3.1

Simplify each term.

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Step 2.2.1.1.3.1.1

Multiply $\frac{3 x}{4} \cdot \frac{3 x}{4}$ .

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Step 2.2.1.1.3.1.1.1

Multiply $\frac{3 x}{4}$ by $\frac{3 x}{4}$ .

$x^{2} + \frac{3 x (3 x)}{4 \cdot 4} + \frac{3 x}{4} \cdot (- \frac{25}{4}) - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.1.2

Multiply $3$ by $3$ .

$x^{2} + \frac{9 x \cdot x}{4 \cdot 4} + \frac{3 x}{4} \cdot (- \frac{25}{4}) - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.1.3

Raise $x$ to the power of $1$ .

$x^{2} + \frac{9 (x \cdot x)}{4 \cdot 4} + \frac{3 x}{4} \cdot (- \frac{25}{4}) - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.1.4

Raise $x$ to the power of $1$ .

$x^{2} + \frac{9 (x \cdot x)}{4 \cdot 4} + \frac{3 x}{4} \cdot (- \frac{25}{4}) - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.1.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} + \frac{9 x^{1 + 1}}{4 \cdot 4} + \frac{3 x}{4} \cdot (- \frac{25}{4}) - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.1.6

Add $1$ and $1$ .

$x^{2} + \frac{9 x^{2}}{4 \cdot 4} + \frac{3 x}{4} \cdot (- \frac{25}{4}) - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.1.7

Multiply $4$ by $4$ .

$x^{2} + \frac{9 x^{2}}{16} + \frac{3 x}{4} \cdot (- \frac{25}{4}) - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$x^{2} + \frac{9 x^{2}}{16} + \frac{3 x}{4} \cdot (- \frac{25}{4}) - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.2

Multiply $\frac{3 x}{4} (- \frac{25}{4})$ .

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Step 2.2.1.1.3.1.2.1

Multiply $\frac{3 x}{4}$ by $\frac{25}{4}$ .

$x^{2} + \frac{9 x^{2}}{16} - \frac{3 x \cdot 25}{4 \cdot 4} - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.2.2

Multiply $25$ by $3$ .

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{4 \cdot 4} - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.2.3

Multiply $4$ by $4$ .

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{16} - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{16} - \frac{25}{4} \cdot \frac{3 x}{4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.3

Multiply $- \frac{25}{4} \cdot \frac{3 x}{4}$ .

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Step 2.2.1.1.3.1.3.1

Multiply $\frac{3 x}{4}$ by $\frac{25}{4}$ .

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{16} - \frac{3 x \cdot 25}{4 \cdot 4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.3.2

Multiply $25$ by $3$ .

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{16} - \frac{75 x}{4 \cdot 4} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.3.3

Multiply $4$ by $4$ .

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{16} - \frac{75 x}{16} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{16} - \frac{75 x}{16} - \frac{25}{4} \cdot (- \frac{25}{4}) = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.4

Multiply $- \frac{25}{4} (- \frac{25}{4})$ .

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Step 2.2.1.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{16} - \frac{75 x}{16} + 1 (\frac{25}{4}) \cdot \frac{25}{4} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.4.2

Multiply $\frac{25}{4}$ by $1$ .

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{16} - \frac{75 x}{16} + \frac{25}{4} \cdot \frac{25}{4} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.4.3

Multiply $\frac{25}{4}$ by $\frac{25}{4}$ .

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{16} - \frac{75 x}{16} + \frac{25 \cdot 25}{4 \cdot 4} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.4.4

Multiply $25$ by $25$ .

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{16} - \frac{75 x}{16} + \frac{625}{4 \cdot 4} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.1.4.5

Multiply $4$ by $4$ .

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{16} - \frac{75 x}{16} + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{16} - \frac{75 x}{16} + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{16} - \frac{75 x}{16} + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.3.2

Subtract $\frac{75 x}{16}$ from $- \frac{75 x}{16}$ .

$x^{2} + \frac{9 x^{2}}{16} - 2 \frac{75 x}{16} + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$x^{2} + \frac{9 x^{2}}{16} - 2 \frac{75 x}{16} + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.4

Simplify each term.

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Step 2.2.1.1.4.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1.1

Factor $2$ out of $- 2$ .

$x^{2} + \frac{9 x^{2}}{16} + 2 (- 1) (\frac{75 x}{16}) + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.4.1.2

Factor $2$ out of $16$ .

$x^{2} + \frac{9 x^{2}}{16} + 2 \cdot (- 1 \frac{75 x}{2 \cdot 8}) + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.4.1.3

Cancel the common factor.

$x^{2} + \frac{9 x^{2}}{16} + 2 \cdot (- 1 \frac{75 x}{2 \cdot 8}) + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.4.1.4

Rewrite the expression.

$x^{2} + \frac{9 x^{2}}{16} - 1 \frac{75 x}{8} + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$x^{2} + \frac{9 x^{2}}{16} - 1 \frac{75 x}{8} + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.1.4.2

Rewrite $- 1 \frac{75 x}{8}$ as $- \frac{75 x}{8}$ .

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{8} + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{8} + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$x^{2} + \frac{9 x^{2}}{16} - \frac{75 x}{8} + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.2

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$x^{2} \cdot \frac{16}{16} + \frac{9 x^{2}}{16} - \frac{75 x}{8} + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $x^{2}$ and $\frac{16}{16}$ .

$\frac{x^{2} \cdot 16}{16} + \frac{9 x^{2}}{16} - \frac{75 x}{8} + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 16 + 9 x^{2}}{16} - \frac{75 x}{8} + \frac{625}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.3.3

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 16 + 9 x^{2} + 625}{16} + \frac{- 75 x}{8} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$\frac{x^{2} \cdot 16 + 9 x^{2} + 625}{16} + \frac{- 75 x}{8} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.4

Move $16$ to the left of $x^{2}$ .

$\frac{16 x^{2} + 9 x^{2} + 625}{16} + \frac{- 75 x}{8} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.5

Add $16 x^{2}$ and $9 x^{2}$ .

$\frac{25 x^{2} + 625}{16} + \frac{- 75 x}{8} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.6

Simplify each term.

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Step 2.2.1.6.1

Factor $25$ out of $25 x^{2} + 625$ .

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Step 2.2.1.6.1.1

Factor $25$ out of $25 x^{2}$ .

$\frac{25 (x^{2}) + 625}{16} + \frac{- 75 x}{8} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.6.1.2

Factor $25$ out of $625$ .

$\frac{25 x^{2} + 25 \cdot 25}{16} + \frac{- 75 x}{8} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.6.1.3

Factor $25$ out of $25 x^{2} + 25 \cdot 25$ .

$\frac{25 (x^{2} + 25)}{16} + \frac{- 75 x}{8} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$\frac{25 (x^{2} + 25)}{16} + \frac{- 75 x}{8} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.6.2

Move the negative in front of the fraction.

$\frac{25 (x^{2} + 25)}{16} - \frac{75 x}{8} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$\frac{25 (x^{2} + 25)}{16} - \frac{75 x}{8} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.7

To write $- \frac{75 x}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{25 (x^{2} + 25)}{16} - \frac{75 x}{8} \cdot \frac{2}{2} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.8

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.8.1

Multiply $\frac{75 x}{8}$ by $\frac{2}{2}$ .

$\frac{25 (x^{2} + 25)}{16} - \frac{75 x \cdot 2}{8 \cdot 2} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.8.2

Multiply $8$ by $2$ .

$\frac{25 (x^{2} + 25)}{16} - \frac{75 x \cdot 2}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$\frac{25 (x^{2} + 25)}{16} - \frac{75 x \cdot 2}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.9

Combine the numerators over the common denominator.

$\frac{25 (x^{2} + 25) - 75 x \cdot 2}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.10

Simplify the numerator.

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Step 2.2.1.10.1

Factor $25$ out of $25 (x^{2} + 25) - 75 x \cdot 2$ .

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Step 2.2.1.10.1.1

Factor $25$ out of $- 75 x \cdot 2$ .

$\frac{25 (x^{2} + 25) + 25 (- 3 x \cdot 2)}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.10.1.2

Factor $25$ out of $25 (x^{2} + 25) + 25 (- 3 x \cdot 2)$ .

$\frac{25 (x^{2} + 25 - 3 x \cdot 2)}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$\frac{25 (x^{2} + 25 - 3 x \cdot 2)}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.10.2

Multiply $2$ by $- 3$ .

$\frac{25 (x^{2} + 25 - 6 x)}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 2.2.1.10.3

Reorder terms.

$\frac{25 (x^{2} - 6 x + 25)}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$\frac{25 (x^{2} - 6 x + 25)}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$\frac{25 (x^{2} - 6 x + 25)}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$\frac{25 (x^{2} - 6 x + 25)}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

$\frac{25 (x^{2} - 6 x + 25)}{16} = 25$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3

Solve for $x$ in $\frac{25 (x^{2} - 6 x + 25)}{16} = 25$ .

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Step 3.1

Multiply both sides by $16$ .

$\frac{25 (x^{2} - 6 x + 25)}{16} \cdot 16 = 25 \cdot 16$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{25 (x^{2} - 6 x + 25)}{16} \cdot 16$ .

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Step 3.2.1.1.1

Cancel the common factor of $16$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{25 (x^{2} - 6 x + 25)}{16} \cdot 16 = 25 \cdot 16$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.2.1.1.1.2

Rewrite the expression.

$25 (x^{2} - 6 x + 25) = 25 \cdot 16$

$y = \frac{3 x}{4} - \frac{25}{4}$

$25 (x^{2} - 6 x + 25) = 25 \cdot 16$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.2.1.1.2

Apply the distributive property.

$25 x^{2} + 25 (- 6 x) + 25 \cdot 25 = 25 \cdot 16$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.2.1.1.3

Simplify.

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Step 3.2.1.1.3.1

Multiply $- 6$ by $25$ .

$25 x^{2} - 150 x + 25 \cdot 25 = 25 \cdot 16$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.2.1.1.3.2

Multiply $25$ by $25$ .

$25 x^{2} - 150 x + 625 = 25 \cdot 16$

$y = \frac{3 x}{4} - \frac{25}{4}$

$25 x^{2} - 150 x + 625 = 25 \cdot 16$

$y = \frac{3 x}{4} - \frac{25}{4}$

$25 x^{2} - 150 x + 625 = 25 \cdot 16$

$y = \frac{3 x}{4} - \frac{25}{4}$

$25 x^{2} - 150 x + 625 = 25 \cdot 16$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $25$ by $16$ .

$25 x^{2} - 150 x + 625 = 400$

$y = \frac{3 x}{4} - \frac{25}{4}$

$25 x^{2} - 150 x + 625 = 400$

$y = \frac{3 x}{4} - \frac{25}{4}$

$25 x^{2} - 150 x + 625 = 400$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3

Solve for $x$ .

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Step 3.3.1

Subtract $400$ from both sides of the equation.

$25 x^{2} - 150 x + 625 - 400 = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.2

Subtract $400$ from $625$ .

$25 x^{2} - 150 x + 225 = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.3

Factor the left side of the equation.

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Step 3.3.3.1

Factor $25$ out of $25 x^{2} - 150 x + 225$ .

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Step 3.3.3.1.1

Factor $25$ out of $25 x^{2}$ .

$25 (x^{2}) - 150 x + 225 = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.3.1.2

Factor $25$ out of $- 150 x$ .

$25 (x^{2}) + 25 (- 6 x) + 225 = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.3.1.3

Factor $25$ out of $225$ .

$25 x^{2} + 25 (- 6 x) + 25 \cdot 9 = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.3.1.4

Factor $25$ out of $25 x^{2} + 25 (- 6 x)$ .

$25 (x^{2} - 6 x) + 25 \cdot 9 = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.3.1.5

Factor $25$ out of $25 (x^{2} - 6 x) + 25 \cdot 9$ .

$25 (x^{2} - 6 x + 9) = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

$25 (x^{2} - 6 x + 9) = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.3.2

Factor using the perfect square rule.

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Step 3.3.3.2.1

Rewrite $9$ as $3^{2}$ .

$25 (x^{2} - 6 x + 3^{2}) = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.3.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.3.2.3

Rewrite the polynomial.

$25 (x^{2} - 2 \cdot x \cdot 3 + 3^{2}) = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.3.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

$25 {(x - 3)}^{2} = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

$25 {(x - 3)}^{2} = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

$25 {(x - 3)}^{2} = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.4

Divide each term in $25 {(x - 3)}^{2} = 0$ by $25$ and simplify.

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Step 3.3.4.1

Divide each term in $25 {(x - 3)}^{2} = 0$ by $25$ .

$\frac{25 {(x - 3)}^{2}}{25} = \frac{0}{25}$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.4.2

Simplify the left side.

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Step 3.3.4.2.1

Cancel the common factor of $25$ .

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Step 3.3.4.2.1.1

Cancel the common factor.

$\frac{25 {(x - 3)}^{2}}{25} = \frac{0}{25}$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.4.2.1.2

Divide ${(x - 3)}^{2}$ by $1$ .

${(x - 3)}^{2} = \frac{0}{25}$

$y = \frac{3 x}{4} - \frac{25}{4}$

${(x - 3)}^{2} = \frac{0}{25}$

$y = \frac{3 x}{4} - \frac{25}{4}$

${(x - 3)}^{2} = \frac{0}{25}$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.4.3

Simplify the right side.

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Step 3.3.4.3.1

Divide $0$ by $25$ .

${(x - 3)}^{2} = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

${(x - 3)}^{2} = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

${(x - 3)}^{2} = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.5

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 3.3.6

Add $3$ to both sides of the equation.

$x = 3$

$y = \frac{3 x}{4} - \frac{25}{4}$

$x = 3$

$y = \frac{3 x}{4} - \frac{25}{4}$

$x = 3$

$y = \frac{3 x}{4} - \frac{25}{4}$

Step 4

Replace all occurrences of $x$ with $3$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = \frac{3 x}{4} - \frac{25}{4}$ with $3$ .

$y = \frac{3 (3)}{4} - \frac{25}{4}$

$x = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3 (3)}{4} - \frac{25}{4}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$y = \frac{3 (3) - 25}{4}$

$x = 3$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $3$ by $3$ .

$y = \frac{9 - 25}{4}$

$x = 3$

Step 4.2.1.2.2

Subtract $25$ from $9$ .

$y = \frac{- 16}{4}$

$x = 3$

Step 4.2.1.2.3

Divide $- 16$ by $4$ .

$y = - 4$

$x = 3$

$y = - 4$

$x = 3$

$y = - 4$

$x = 3$

$y = - 4$

$x = 3$

$y = - 4$

$x = 3$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(3, - 4)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(3, - 4)$

Equation Form:

$x = 3, y = - 4$

Step 7