Precalculus Examples

$3 x + 4 y = 16$ , $- 4 x - 3 y = - 19$

Step 1

Solve for $x$ in $3 x + 4 y = 16$ .

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Step 1.1

Subtract $4 y$ from both sides of the equation.

$3 x = 16 - 4 y$

$- 4 x - 3 y = - 19$

Step 1.2

Divide each term in $3 x = 16 - 4 y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 16 - 4 y$ by $3$ .

$\frac{3 x}{3} = \frac{16}{3} + \frac{- 4 y}{3}$

$- 4 x - 3 y = - 19$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{16}{3} + \frac{- 4 y}{3}$

$- 4 x - 3 y = - 19$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{16}{3} + \frac{- 4 y}{3}$

$- 4 x - 3 y = - 19$

$x = \frac{16}{3} + \frac{- 4 y}{3}$

$- 4 x - 3 y = - 19$

$x = \frac{16}{3} + \frac{- 4 y}{3}$

$- 4 x - 3 y = - 19$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x = \frac{16}{3} - \frac{4 y}{3}$

$- 4 x - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

$- 4 x - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

$- 4 x - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

$- 4 x - 3 y = - 19$

Step 2

Replace all occurrences of $x$ with $\frac{16}{3} - \frac{4 y}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 4 x - 3 y = - 19$ with $\frac{16}{3} - \frac{4 y}{3}$ .

$- 4 (\frac{16}{3} - \frac{4 y}{3}) - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 4 (\frac{16}{3} - \frac{4 y}{3}) - 3 y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$- 4 (\frac{16}{3}) - 4 (- \frac{4 y}{3}) - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.1.2

Multiply $- 4 (\frac{16}{3})$ .

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Step 2.2.1.1.2.1

Combine $- 4$ and $\frac{16}{3}$ .

$\frac{- 4 \cdot 16}{3} - 4 (- \frac{4 y}{3}) - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.1.2.2

Multiply $- 4$ by $16$ .

$\frac{- 64}{3} - 4 (- \frac{4 y}{3}) - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

$\frac{- 64}{3} - 4 (- \frac{4 y}{3}) - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.1.3

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

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Step 2.2.1.1.3.1

Multiply $- 1$ by $- 4$ .

$\frac{- 64}{3} + 4 (\frac{4 y}{3}) - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.1.3.2

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

$\frac{- 64}{3} + \frac{4 (4 y)}{3} - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.1.3.3

Multiply $4$ by $4$ .

$\frac{- 64}{3} + \frac{16 y}{3} - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

$\frac{- 64}{3} + \frac{16 y}{3} - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.1.4

Move the negative in front of the fraction.

$- \frac{64}{3} + \frac{16 y}{3} - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

$- \frac{64}{3} + \frac{16 y}{3} - 3 y = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.2

To write $- 3 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{64}{3} + \frac{16 y}{3} - 3 y \cdot \frac{3}{3} = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.3

Combine $- 3 y$ and $\frac{3}{3}$ .

$- \frac{64}{3} + \frac{16 y}{3} + \frac{- 3 y \cdot 3}{3} = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{64}{3} + \frac{16 y - 3 y \cdot 3}{3} = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 64 + 16 y - 3 y \cdot 3}{3} = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.6

Multiply $3$ by $- 3$ .

$\frac{- 64 + 16 y - 9 y}{3} = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.7

Subtract $9 y$ from $16 y$ .

$\frac{- 64 + 7 y}{3} = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.8

Rewrite $- 64$ as $- 1 (64)$ .

$\frac{- 1 \cdot 64 + 7 y}{3} = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.9

Factor $- 1$ out of $7 y$ .

$\frac{- 1 \cdot 64 - (- 7 y)}{3} = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.10

Factor $- 1$ out of $- 1 (64) - (- 7 y)$ .

$\frac{- 1 (64 - 7 y)}{3} = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 2.2.1.11

Move the negative in front of the fraction.

$- \frac{64 - 7 y}{3} = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

$- \frac{64 - 7 y}{3} = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

$- \frac{64 - 7 y}{3} = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

$- \frac{64 - 7 y}{3} = - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3

Solve for $y$ in $- \frac{64 - 7 y}{3} = - 19$ .

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Step 3.1

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{64 - 7 y}{3}) = - 3 \cdot - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $- 3 (- \frac{64 - 7 y}{3})$ .

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Step 3.2.1.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{64 - 7 y}{3}$ into the numerator.

$- 3 \frac{- (64 - 7 y)}{3} = - 3 \cdot - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3.2.1.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- (64 - 7 y)}{3}) = - 3 \cdot - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3.2.1.1.1.3

Cancel the common factor.

$3 \cdot (- 1 \frac{- (64 - 7 y)}{3}) = - 3 \cdot - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3.2.1.1.1.4

Rewrite the expression.

$64 - 7 y = - 3 \cdot - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

$64 - 7 y = - 3 \cdot - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (64 - 7 y) = - 3 \cdot - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3.2.1.1.2.2

Multiply $64 - 7 y$ by $1$ .

$64 - 7 y = - 3 \cdot - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

$64 - 7 y = - 3 \cdot - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

$64 - 7 y = - 3 \cdot - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

$64 - 7 y = - 3 \cdot - 19$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 3$ by $- 19$ .

$64 - 7 y = 57$

$x = \frac{16}{3} - \frac{4 y}{3}$

$64 - 7 y = 57$

$x = \frac{16}{3} - \frac{4 y}{3}$

$64 - 7 y = 57$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3.3

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

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Step 3.3.1

Subtract $64$ from both sides of the equation.

$- 7 y = 57 - 64$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3.3.2

Subtract $64$ from $57$ .

$- 7 y = - 7$

$x = \frac{16}{3} - \frac{4 y}{3}$

$- 7 y = - 7$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3.4

Divide each term in $- 7 y = - 7$ by $- 7$ and simplify.

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Step 3.4.1

Divide each term in $- 7 y = - 7$ by $- 7$ .

$\frac{- 7 y}{- 7} = \frac{- 7}{- 7}$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $- 7$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 7 y}{- 7} = \frac{- 7}{- 7}$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 7}{- 7}$

$x = \frac{16}{3} - \frac{4 y}{3}$

$y = \frac{- 7}{- 7}$

$x = \frac{16}{3} - \frac{4 y}{3}$

$y = \frac{- 7}{- 7}$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $- 7$ by $- 7$ .

$y = 1$

$x = \frac{16}{3} - \frac{4 y}{3}$

$y = 1$

$x = \frac{16}{3} - \frac{4 y}{3}$

$y = 1$

$x = \frac{16}{3} - \frac{4 y}{3}$

$y = 1$

$x = \frac{16}{3} - \frac{4 y}{3}$

Step 4

Replace all occurrences of $y$ with $1$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{16}{3} - \frac{4 y}{3}$ with $1$ .

$x = \frac{16}{3} - \frac{4 (1)}{3}$

$y = 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{16}{3} - \frac{4 (1)}{3}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{16 - 4 \cdot 1}{3}$

$y = 1$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{16 - 4}{3}$

$y = 1$

Step 4.2.1.2.2

Subtract $4$ from $16$ .

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

$y = 1$

Step 4.2.1.2.3

Divide $12$ by $3$ .

$x = 4$

$y = 1$

$x = 4$

$y = 1$

$x = 4$

$y = 1$

$x = 4$

$y = 1$

$x = 4$

$y = 1$

Step 5

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

$(4, 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(4, 1)$

Equation Form:

$x = 4, y = 1$

Step 7