Precalculus Examples

$9 x^{2} + 16 y^{2} = 144$ , $4 y + 3 x = 12$

Step 1

Solve for $y$ in $4 y + 3 x = 12$ .

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Step 1.1

Subtract $3 x$ from both sides of the equation.

$4 y = 12 - 3 x$

$9 x^{2} + 16 y^{2} = 144$

Step 1.2

Divide each term in $4 y = 12 - 3 x$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 y = 12 - 3 x$ by $4$ .

$\frac{4 y}{4} = \frac{12}{4} + \frac{- 3 x}{4}$

$9 x^{2} + 16 y^{2} = 144$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{12}{4} + \frac{- 3 x}{4}$

$9 x^{2} + 16 y^{2} = 144$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{12}{4} + \frac{- 3 x}{4}$

$9 x^{2} + 16 y^{2} = 144$

$y = \frac{12}{4} + \frac{- 3 x}{4}$

$9 x^{2} + 16 y^{2} = 144$

$y = \frac{12}{4} + \frac{- 3 x}{4}$

$9 x^{2} + 16 y^{2} = 144$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $12$ by $4$ .

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

$9 x^{2} + 16 y^{2} = 144$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$9 x^{2} + 16 y^{2} = 144$

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

$9 x^{2} + 16 y^{2} = 144$

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

$9 x^{2} + 16 y^{2} = 144$

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

$9 x^{2} + 16 y^{2} = 144$

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

$9 x^{2} + 16 y^{2} = 144$

Step 2

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

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Step 2.1

Replace all occurrences of $y$ in $9 x^{2} + 16 y^{2} = 144$ with $3 - \frac{3 x}{4}$ .

$9 x^{2} + 16 {(3 - \frac{3 x}{4})}^{2} = 144$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $9 x^{2} + 16 {(3 - \frac{3 x}{4})}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

$9 x^{2} + 16 ((3 - \frac{3 x}{4}) (3 - \frac{3 x}{4})) = 144$

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

Step 2.2.1.1.2

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

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Step 2.2.1.1.2.1

Apply the distributive property.

$9 x^{2} + 16 (3 (3 - \frac{3 x}{4}) - \frac{3 x}{4} \cdot (3 - \frac{3 x}{4})) = 144$

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

Step 2.2.1.1.2.2

Apply the distributive property.

$9 x^{2} + 16 (3 \cdot 3 + 3 (- \frac{3 x}{4}) - \frac{3 x}{4} \cdot (3 - \frac{3 x}{4})) = 144$

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

Step 2.2.1.1.2.3

Apply the distributive property.

$9 x^{2} + 16 (3 \cdot 3 + 3 (- \frac{3 x}{4}) - \frac{3 x}{4} \cdot 3 - \frac{3 x}{4} \cdot (- \frac{3 x}{4})) = 144$

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

$9 x^{2} + 16 (3 \cdot 3 + 3 (- \frac{3 x}{4}) - \frac{3 x}{4} \cdot 3 - \frac{3 x}{4} \cdot (- \frac{3 x}{4})) = 144$

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

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 $3$ by $3$ .

$9 x^{2} + 16 (9 + 3 (- \frac{3 x}{4}) - \frac{3 x}{4} \cdot 3 - \frac{3 x}{4} \cdot (- \frac{3 x}{4})) = 144$

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

Step 2.2.1.1.3.1.2

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

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Step 2.2.1.1.3.1.2.1

Multiply $- 1$ by $3$ .

$9 x^{2} + 16 (9 - 3 \frac{3 x}{4} - \frac{3 x}{4} \cdot 3 - \frac{3 x}{4} \cdot (- \frac{3 x}{4})) = 144$

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

Step 2.2.1.1.3.1.2.2

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

$9 x^{2} + 16 (9 + \frac{- 3 (3 x)}{4} - \frac{3 x}{4} \cdot 3 - \frac{3 x}{4} \cdot (- \frac{3 x}{4})) = 144$

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

Step 2.2.1.1.3.1.2.3

Multiply $3$ by $- 3$ .

$9 x^{2} + 16 (9 + \frac{- 9 x}{4} - \frac{3 x}{4} \cdot 3 - \frac{3 x}{4} \cdot (- \frac{3 x}{4})) = 144$

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

$9 x^{2} + 16 (9 + \frac{- 9 x}{4} - \frac{3 x}{4} \cdot 3 - \frac{3 x}{4} \cdot (- \frac{3 x}{4})) = 144$

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

Step 2.2.1.1.3.1.3

Move the negative in front of the fraction.

$9 x^{2} + 16 (9 - \frac{9 x}{4} - \frac{3 x}{4} \cdot 3 - \frac{3 x}{4} \cdot (- \frac{3 x}{4})) = 144$

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

Step 2.2.1.1.3.1.4

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

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Step 2.2.1.1.3.1.4.1

Multiply $3$ by $- 1$ .

$9 x^{2} + 16 (9 - \frac{9 x}{4} - 3 \frac{3 x}{4} - \frac{3 x}{4} \cdot (- \frac{3 x}{4})) = 144$

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

Step 2.2.1.1.3.1.4.2

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

$9 x^{2} + 16 (9 - \frac{9 x}{4} + \frac{- 3 (3 x)}{4} - \frac{3 x}{4} \cdot (- \frac{3 x}{4})) = 144$

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

Step 2.2.1.1.3.1.4.3

Multiply $3$ by $- 3$ .

$9 x^{2} + 16 (9 - \frac{9 x}{4} + \frac{- 9 x}{4} - \frac{3 x}{4} \cdot (- \frac{3 x}{4})) = 144$

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

$9 x^{2} + 16 (9 - \frac{9 x}{4} + \frac{- 9 x}{4} - \frac{3 x}{4} \cdot (- \frac{3 x}{4})) = 144$

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

Step 2.2.1.1.3.1.5

Move the negative in front of the fraction.

$9 x^{2} + 16 (9 - \frac{9 x}{4} - \frac{9 x}{4} - \frac{3 x}{4} \cdot (- \frac{3 x}{4})) = 144$

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

Step 2.2.1.1.3.1.6

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

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Step 2.2.1.1.3.1.6.1

Multiply $- 1$ by $- 1$ .

$9 x^{2} + 16 (9 - \frac{9 x}{4} - \frac{9 x}{4} + 1 (\frac{3 x}{4}) \cdot \frac{3 x}{4}) = 144$

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

Step 2.2.1.1.3.1.6.2

Multiply $\frac{3 x}{4}$ by $1$ .

$9 x^{2} + 16 (9 - \frac{9 x}{4} - \frac{9 x}{4} + \frac{3 x}{4} \cdot \frac{3 x}{4}) = 144$

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

Step 2.2.1.1.3.1.6.3

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

$9 x^{2} + 16 (9 - \frac{9 x}{4} - \frac{9 x}{4} + \frac{3 x (3 x)}{4 \cdot 4}) = 144$

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

Step 2.2.1.1.3.1.6.4

Multiply $3$ by $3$ .

$9 x^{2} + 16 (9 - \frac{9 x}{4} - \frac{9 x}{4} + \frac{9 x \cdot x}{4 \cdot 4}) = 144$

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

Step 2.2.1.1.3.1.6.5

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

$9 x^{2} + 16 (9 - \frac{9 x}{4} - \frac{9 x}{4} + \frac{9 (x \cdot x)}{4 \cdot 4}) = 144$

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

Step 2.2.1.1.3.1.6.6

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

$9 x^{2} + 16 (9 - \frac{9 x}{4} - \frac{9 x}{4} + \frac{9 (x \cdot x)}{4 \cdot 4}) = 144$

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

Step 2.2.1.1.3.1.6.7

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

$9 x^{2} + 16 (9 - \frac{9 x}{4} - \frac{9 x}{4} + \frac{9 x^{1 + 1}}{4 \cdot 4}) = 144$

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

Step 2.2.1.1.3.1.6.8

Add $1$ and $1$ .

$9 x^{2} + 16 (9 - \frac{9 x}{4} - \frac{9 x}{4} + \frac{9 x^{2}}{4 \cdot 4}) = 144$

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

Step 2.2.1.1.3.1.6.9

Multiply $4$ by $4$ .

$9 x^{2} + 16 (9 - \frac{9 x}{4} - \frac{9 x}{4} + \frac{9 x^{2}}{16}) = 144$

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

$9 x^{2} + 16 (9 - \frac{9 x}{4} - \frac{9 x}{4} + \frac{9 x^{2}}{16}) = 144$

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

$9 x^{2} + 16 (9 - \frac{9 x}{4} - \frac{9 x}{4} + \frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.3.2

Subtract $\frac{9 x}{4}$ from $- \frac{9 x}{4}$ .

$9 x^{2} + 16 (9 - 2 \frac{9 x}{4} + \frac{9 x^{2}}{16}) = 144$

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

$9 x^{2} + 16 (9 - 2 \frac{9 x}{4} + \frac{9 x^{2}}{16}) = 144$

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

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 $- 2$ .

$9 x^{2} + 16 (9 + 2 (- 1) (\frac{9 x}{4}) + \frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.4.1.2

Factor $2$ out of $4$ .

$9 x^{2} + 16 (9 + 2 \cdot (- 1 \frac{9 x}{2 \cdot 2}) + \frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.4.1.3

Cancel the common factor.

$9 x^{2} + 16 (9 + 2 \cdot (- 1 \frac{9 x}{2 \cdot 2}) + \frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.4.1.4

Rewrite the expression.

$9 x^{2} + 16 (9 - 1 \frac{9 x}{2} + \frac{9 x^{2}}{16}) = 144$

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

$9 x^{2} + 16 (9 - 1 \frac{9 x}{2} + \frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.4.2

Rewrite $- 1 \frac{9 x}{2}$ as $- \frac{9 x}{2}$ .

$9 x^{2} + 16 (9 - \frac{9 x}{2} + \frac{9 x^{2}}{16}) = 144$

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

$9 x^{2} + 16 (9 - \frac{9 x}{2} + \frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.5

Apply the distributive property.

$9 x^{2} + 16 \cdot 9 + 16 (- \frac{9 x}{2}) + 16 (\frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.6

Simplify.

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Step 2.2.1.1.6.1

Multiply $16$ by $9$ .

$9 x^{2} + 144 + 16 (- \frac{9 x}{2}) + 16 (\frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.6.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.6.2.1

Move the leading negative in $- \frac{9 x}{2}$ into the numerator.

$9 x^{2} + 144 + 16 (\frac{- 9 x}{2}) + 16 (\frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.6.2.2

Factor $2$ out of $16$ .

$9 x^{2} + 144 + 2 (8) (\frac{- 9 x}{2}) + 16 (\frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.6.2.3

Cancel the common factor.

$9 x^{2} + 144 + 2 \cdot (8 (\frac{- 9 x}{2})) + 16 (\frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.6.2.4

Rewrite the expression.

$9 x^{2} + 144 + 8 (- 9 x) + 16 (\frac{9 x^{2}}{16}) = 144$

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

$9 x^{2} + 144 + 8 (- 9 x) + 16 (\frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.6.3

Multiply $- 9$ by $8$ .

$9 x^{2} + 144 - 72 x + 16 (\frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.6.4

Cancel the common factor of $16$ .

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Step 2.2.1.1.6.4.1

Cancel the common factor.

$9 x^{2} + 144 - 72 x + 16 (\frac{9 x^{2}}{16}) = 144$

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

Step 2.2.1.1.6.4.2

Rewrite the expression.

$9 x^{2} + 144 - 72 x + 9 x^{2} = 144$

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

$9 x^{2} + 144 - 72 x + 9 x^{2} = 144$

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

$9 x^{2} + 144 - 72 x + 9 x^{2} = 144$

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

$9 x^{2} + 144 - 72 x + 9 x^{2} = 144$

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

Step 2.2.1.2

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

$18 x^{2} + 144 - 72 x = 144$

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

$18 x^{2} + 144 - 72 x = 144$

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

$18 x^{2} + 144 - 72 x = 144$

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

$18 x^{2} + 144 - 72 x = 144$

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

Step 3

Solve for $x$ in $18 x^{2} + 144 - 72 x = 144$ .

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Step 3.1

Subtract $144$ from both sides of the equation.

$18 x^{2} + 144 - 72 x - 144 = 0$

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

Step 3.2

Combine the opposite terms in $18 x^{2} + 144 - 72 x - 144$ .

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Step 3.2.1

Subtract $144$ from $144$ .

$18 x^{2} - 72 x + 0 = 0$

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

Step 3.2.2

Add $18 x^{2} - 72 x$ and $0$ .

$18 x^{2} - 72 x = 0$

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

$18 x^{2} - 72 x = 0$

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

Step 3.3

Factor $18 x$ out of $18 x^{2} - 72 x$ .

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Step 3.3.1

Factor $18 x$ out of $18 x^{2}$ .

$18 x (x) - 72 x = 0$

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

Step 3.3.2

Factor $18 x$ out of $- 72 x$ .

$18 x (x) + 18 x (- 4) = 0$

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

Step 3.3.3

Factor $18 x$ out of $18 x (x) + 18 x (- 4)$ .

$18 x (x - 4) = 0$

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

$18 x (x - 4) = 0$

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

Step 3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 4 = 0$

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

Step 3.5

Set $x$ equal to $0$ .

$x = 0$

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

Step 3.6

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

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

Step 3.6.2

Add $4$ to both sides of the equation.

$x = 4$

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

$x = 4$

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

Step 3.7

The final solution is all the values that make $18 x (x - 4) = 0$ true.

$x = 0, 4$

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

$x = 0, 4$

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

Step 4

Replace all occurrences of $x$ with $0$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = 3 - \frac{3 x}{4}$ with $0$ .

$y = 3 - \frac{3 (0)}{4}$

$x = 0$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $3 - \frac{3 (0)}{4}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $0$ and $4$ .

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Step 4.2.1.1.1.1

Factor $4$ out of $3 (0)$ .

$y = 3 - \frac{4 (3 \cdot (0))}{4}$

$x = 0$

Step 4.2.1.1.1.2

Cancel the common factors.

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Step 4.2.1.1.1.2.1

Factor $4$ out of $4$ .

$y = 3 - \frac{4 (3 \cdot (0))}{4 (1)}$

$x = 0$

Step 4.2.1.1.1.2.2

Cancel the common factor.

$y = 3 - \frac{4 (3 \cdot (0))}{4 \cdot 1}$

$x = 0$

Step 4.2.1.1.1.2.3

Rewrite the expression.

$y = 3 - \frac{3 \cdot (0)}{1}$

$x = 0$

Step 4.2.1.1.1.2.4

Divide $3 \cdot (0)$ by $1$ .

$y = 3 - (3 \cdot (0))$

$x = 0$

$y = 3 - (3 \cdot (0))$

$x = 0$

$y = 3 - (3 \cdot (0))$

$x = 0$

Step 4.2.1.1.2

Multiply $- (3 \cdot (0))$ .

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Step 4.2.1.1.2.1

Multiply $3$ by $0$ .

$y = 3 - 0$

$x = 0$

Step 4.2.1.1.2.2

Multiply $- 1$ by $0$ .

$y = 3 + 0$

$x = 0$

$y = 3 + 0$

$x = 0$

$y = 3 + 0$

$x = 0$

Step 4.2.1.2

Add $3$ and $0$ .

$y = 3$

$x = 0$

$y = 3$

$x = 0$

$y = 3$

$x = 0$

$y = 3$

$x = 0$

Step 5

Replace all occurrences of $x$ with $4$ in each equation.

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Step 5.1

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

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

$x = 4$

Step 5.2

Simplify the right side.

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Step 5.2.1

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

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Cancel the common factor of $4$ .

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Step 5.2.1.1.1.1

Cancel the common factor.

$y = 3 - \frac{3 \cdot 4}{4}$

$x = 4$

Step 5.2.1.1.1.2

Divide $3$ by $1$ .

$y = 3 - 1 \cdot 3$

$x = 4$

$y = 3 - 1 \cdot 3$

$x = 4$

Step 5.2.1.1.2

Multiply $- 1$ by $3$ .

$y = 3 - 3$

$x = 4$

$y = 3 - 3$

$x = 4$

Step 5.2.1.2

Subtract $3$ from $3$ .

$y = 0$

$x = 4$

$y = 0$

$x = 4$

$y = 0$

$x = 4$

$y = 0$

$x = 4$

Step 6

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

$(0, 3)$

$(4, 0)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(0, 3), (4, 0)$

Equation Form:

$x = 0, y = 3$

$x = 4, y = 0$

Step 8