Precalculus Examples

$3 x + 2 y = - 5$ , $2 x - 5 y = - 16$

Step 1

Solve for $x$ in $3 x + 2 y = - 5$ .

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Step 1.1

Subtract $2 y$ from both sides of the equation.

$3 x = - 5 - 2 y$

$2 x - 5 y = - 16$

Step 1.2

Divide each term in $3 x = - 5 - 2 y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = - 5 - 2 y$ by $3$ .

$\frac{3 x}{3} = \frac{- 5}{3} + \frac{- 2 y}{3}$

$2 x - 5 y = - 16$

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.

$\frac{3 x}{3} = \frac{- 5}{3} + \frac{- 2 y}{3}$

$2 x - 5 y = - 16$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{3} + \frac{- 2 y}{3}$

$2 x - 5 y = - 16$

$x = \frac{- 5}{3} + \frac{- 2 y}{3}$

$2 x - 5 y = - 16$

$x = \frac{- 5}{3} + \frac{- 2 y}{3}$

$2 x - 5 y = - 16$

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

Move the negative in front of the fraction.

$x = - \frac{5}{3} + \frac{- 2 y}{3}$

$2 x - 5 y = - 16$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{5}{3} - \frac{2 y}{3}$

$2 x - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$2 x - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$2 x - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$2 x - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$2 x - 5 y = - 16$

Step 2

Replace all occurrences of $x$ with $- \frac{5}{3} - \frac{2 y}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x - 5 y = - 16$ with $- \frac{5}{3} - \frac{2 y}{3}$ .

$2 (- \frac{5}{3} - \frac{2 y}{3}) - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (- \frac{5}{3} - \frac{2 y}{3}) - 5 y$ .

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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.

$2 (- \frac{5}{3}) + 2 (- \frac{2 y}{3}) - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.2

Multiply $2 (- \frac{5}{3})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $2$ .

$- 2 (\frac{5}{3}) + 2 (- \frac{2 y}{3}) - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.2.2

Combine $- 2$ and $\frac{5}{3}$ .

$\frac{- 2 \cdot 5}{3} + 2 (- \frac{2 y}{3}) - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.2.3

Multiply $- 2$ by $5$ .

$\frac{- 10}{3} + 2 (- \frac{2 y}{3}) - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$\frac{- 10}{3} + 2 (- \frac{2 y}{3}) - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.3

Multiply $2 (- \frac{2 y}{3})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $2$ .

$\frac{- 10}{3} - 2 \frac{2 y}{3} - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.3.2

Combine $- 2$ and $\frac{2 y}{3}$ .

$\frac{- 10}{3} + \frac{- 2 (2 y)}{3} - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.3.3

Multiply $2$ by $- 2$ .

$\frac{- 10}{3} + \frac{- 4 y}{3} - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$\frac{- 10}{3} + \frac{- 4 y}{3} - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.4

Simplify each term.

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Step 2.2.1.1.4.1

Move the negative in front of the fraction.

$- \frac{10}{3} + \frac{- 4 y}{3} - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{10}{3} - \frac{4 y}{3} - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$- \frac{10}{3} - \frac{4 y}{3} - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$- \frac{10}{3} - \frac{4 y}{3} - 5 y = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.2

To write $- 5 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{10}{3} - \frac{4 y}{3} - 5 y \cdot \frac{3}{3} = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.3

Combine $- 5 y$ and $\frac{3}{3}$ .

$- \frac{10}{3} - \frac{4 y}{3} + \frac{- 5 y \cdot 3}{3} = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{10}{3} + \frac{- 4 y - 5 y \cdot 3}{3} = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 10 - 4 y - 5 y \cdot 3}{3} = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.6

Multiply $3$ by $- 5$ .

$\frac{- 10 - 4 y - 15 y}{3} = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.7

Subtract $15 y$ from $- 4 y$ .

$\frac{- 10 - 19 y}{3} = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.8

Rewrite $- 10$ as $- 1 (10)$ .

$\frac{- 1 \cdot 10 - 19 y}{3} = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.9

Factor $- 1$ out of $- 19 y$ .

$\frac{- 1 \cdot 10 - (19 y)}{3} = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.10

Factor $- 1$ out of $- 1 (10) - (19 y)$ .

$\frac{- 1 (10 + 19 y)}{3} = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.11

Move the negative in front of the fraction.

$- \frac{10 + 19 y}{3} = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$- \frac{10 + 19 y}{3} = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$- \frac{10 + 19 y}{3} = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$- \frac{10 + 19 y}{3} = - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 3

Solve for $y$ in $- \frac{10 + 19 y}{3} = - 16$ .

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Step 3.1

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

$- 3 (- \frac{10 + 19 y}{3}) = - 3 \cdot - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

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 $- 3 (- \frac{10 + 19 y}{3})$ .

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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 $- \frac{10 + 19 y}{3}$ into the numerator.

$- 3 \frac{- (10 + 19 y)}{3} = - 3 \cdot - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 3.2.1.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- (10 + 19 y)}{3}) = - 3 \cdot - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 3.2.1.1.1.3

Cancel the common factor.

$3 \cdot (- 1 \frac{- (10 + 19 y)}{3}) = - 3 \cdot - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 3.2.1.1.1.4

Rewrite the expression.

$10 + 19 y = - 3 \cdot - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$10 + 19 y = - 3 \cdot - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (10 + 19 y) = - 3 \cdot - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 3.2.1.1.2.2

Multiply $10 + 19 y$ by $1$ .

$10 + 19 y = - 3 \cdot - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$10 + 19 y = - 3 \cdot - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$10 + 19 y = - 3 \cdot - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$10 + 19 y = - 3 \cdot - 16$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 3$ by $- 16$ .

$10 + 19 y = 48$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$10 + 19 y = 48$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$10 + 19 y = 48$

$x = - \frac{5}{3} - \frac{2 y}{3}$

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 $10$ from both sides of the equation.

$19 y = 48 - 10$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 3.3.2

Subtract $10$ from $48$ .

$19 y = 38$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$19 y = 38$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 3.4

Divide each term in $19 y = 38$ by $19$ and simplify.

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Step 3.4.1

Divide each term in $19 y = 38$ by $19$ .

$\frac{19 y}{19} = \frac{38}{19}$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $19$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{19 y}{19} = \frac{38}{19}$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{38}{19}$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$y = \frac{38}{19}$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$y = \frac{38}{19}$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $38$ by $19$ .

$y = 2$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$y = 2$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$y = 2$

$x = - \frac{5}{3} - \frac{2 y}{3}$

$y = 2$

$x = - \frac{5}{3} - \frac{2 y}{3}$

Step 4

Replace all occurrences of $y$ with $2$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{5}{3} - \frac{2 y}{3}$ with $2$ .

$x = - \frac{5}{3} - \frac{2 (2)}{3}$

$y = 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{5}{3} - \frac{2 (2)}{3}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 5 - 2 \cdot 2}{3}$

$y = 2$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $- 2$ by $2$ .

$x = \frac{- 5 - 4}{3}$

$y = 2$

Step 4.2.1.2.2

Subtract $4$ from $- 5$ .

$x = \frac{- 9}{3}$

$y = 2$

Step 4.2.1.2.3

Divide $- 9$ by $3$ .

$x = - 3$

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

$y = 2$

Step 5

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

$(- 3, 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 3, 2)$

Equation Form:

$x = - 3, y = 2$

Step 7