Precalculus Examples

$3 x - 2 y = 8$ , $5 x + 3 y = 7$

Step 1

Solve for $x$ in $3 x - 2 y = 8$ .

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Step 1.1

Add $2 y$ to both sides of the equation.

$3 x = 8 + 2 y$

$5 x + 3 y = 7$

Step 1.2

Divide each term in $3 x = 8 + 2 y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 8 + 2 y$ by $3$ .

$\frac{3 x}{3} = \frac{8}{3} + \frac{2 y}{3}$

$5 x + 3 y = 7$

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.

$\frac{3 x}{3} = \frac{8}{3} + \frac{2 y}{3}$

$5 x + 3 y = 7$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{3} + \frac{2 y}{3}$

$5 x + 3 y = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

$5 x + 3 y = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

$5 x + 3 y = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

$5 x + 3 y = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

$5 x + 3 y = 7$

Step 2

Replace all occurrences of $x$ with $\frac{8}{3} + \frac{2 y}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $5 x + 3 y = 7$ with $\frac{8}{3} + \frac{2 y}{3}$ .

$5 (\frac{8}{3} + \frac{2 y}{3}) + 3 y = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 (\frac{8}{3} + \frac{2 y}{3}) + 3 y$ .

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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.

$5 (\frac{8}{3}) + 5 (\frac{2 y}{3}) + 3 y = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 2.2.1.1.2

Multiply $5 (\frac{8}{3})$ .

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Step 2.2.1.1.2.1

Combine $5$ and $\frac{8}{3}$ .

$\frac{5 \cdot 8}{3} + 5 (\frac{2 y}{3}) + 3 y = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 2.2.1.1.2.2

Multiply $5$ by $8$ .

$\frac{40}{3} + 5 (\frac{2 y}{3}) + 3 y = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

$\frac{40}{3} + 5 (\frac{2 y}{3}) + 3 y = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 2.2.1.1.3

Multiply $5 \frac{2 y}{3}$ .

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Step 2.2.1.1.3.1

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

$\frac{40}{3} + \frac{5 (2 y)}{3} + 3 y = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 2.2.1.1.3.2

Multiply $2$ by $5$ .

$\frac{40}{3} + \frac{10 y}{3} + 3 y = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

$\frac{40}{3} + \frac{10 y}{3} + 3 y = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

$\frac{40}{3} + \frac{10 y}{3} + 3 y = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 2.2.1.2

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

$\frac{40}{3} + \frac{10 y}{3} + 3 y \cdot \frac{3}{3} = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 2.2.1.3

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

$\frac{40}{3} + \frac{10 y}{3} + \frac{3 y \cdot 3}{3} = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{40}{3} + \frac{10 y + 3 y \cdot 3}{3} = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{40 + 10 y + 3 y \cdot 3}{3} = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 2.2.1.6

Multiply $3$ by $3$ .

$\frac{40 + 10 y + 9 y}{3} = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 2.2.1.7

Add $10 y$ and $9 y$ .

$\frac{40 + 19 y}{3} = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

$\frac{40 + 19 y}{3} = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

$\frac{40 + 19 y}{3} = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

$\frac{40 + 19 y}{3} = 7$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 3

Solve for $y$ in $\frac{40 + 19 y}{3} = 7$ .

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Step 3.1

Multiply both sides by $3$ .

$\frac{40 + 19 y}{3} \cdot 3 = 7 \cdot 3$

$x = \frac{8}{3} + \frac{2 y}{3}$

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 $\frac{40 + 19 y}{3} \cdot 3$ .

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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

Cancel the common factor.

$\frac{40 + 19 y}{3} \cdot 3 = 7 \cdot 3$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 3.2.1.1.1.2

Rewrite the expression.

$40 + 19 y = 7 \cdot 3$

$x = \frac{8}{3} + \frac{2 y}{3}$

$40 + 19 y = 7 \cdot 3$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 3.2.1.1.2

Reorder $40$ and $19 y$ .

$19 y + 40 = 7 \cdot 3$

$x = \frac{8}{3} + \frac{2 y}{3}$

$19 y + 40 = 7 \cdot 3$

$x = \frac{8}{3} + \frac{2 y}{3}$

$19 y + 40 = 7 \cdot 3$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $7$ by $3$ .

$19 y + 40 = 21$

$x = \frac{8}{3} + \frac{2 y}{3}$

$19 y + 40 = 21$

$x = \frac{8}{3} + \frac{2 y}{3}$

$19 y + 40 = 21$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $40$ from both sides of the equation.

$19 y = 21 - 40$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 3.3.1.2

Subtract $40$ from $21$ .

$19 y = - 19$

$x = \frac{8}{3} + \frac{2 y}{3}$

$19 y = - 19$

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 3.3.2

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

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Step 3.3.2.1

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

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

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $19$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

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

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 3.3.2.2.1.2

Divide $y$ by $1$ .

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

$x = \frac{8}{3} + \frac{2 y}{3}$

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

$x = \frac{8}{3} + \frac{2 y}{3}$

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

$x = \frac{8}{3} + \frac{2 y}{3}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $- 19$ by $19$ .

$y = - 1$

$x = \frac{8}{3} + \frac{2 y}{3}$

$y = - 1$

$x = \frac{8}{3} + \frac{2 y}{3}$

$y = - 1$

$x = \frac{8}{3} + \frac{2 y}{3}$

$y = - 1$

$x = \frac{8}{3} + \frac{2 y}{3}$

$y = - 1$

$x = \frac{8}{3} + \frac{2 y}{3}$

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 $x = \frac{8}{3} + \frac{2 y}{3}$ with $- 1$ .

$x = \frac{8}{3} + \frac{2 (- 1)}{3}$

$y = - 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{8}{3} + \frac{2 (- 1)}{3}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{8 + 2 (- 1)}{3}$

$y = - 1$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $2$ by $- 1$ .

$x = \frac{8 - 2}{3}$

$y = - 1$

Step 4.2.1.2.2

Subtract $2$ from $8$ .

$x = \frac{6}{3}$

$y = - 1$

Step 4.2.1.2.3

Divide $6$ by $3$ .

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

Step 5

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

$(2, - 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, - 1)$

Equation Form:

$x = 2, y = - 1$

Step 7