Precalculus Examples

$2 x - 3 y = 3$ , $5 x - 7 y = 9$

Step 1

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

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Step 1.1

Add $3 y$ to both sides of the equation.

$2 x = 3 + 3 y$

$5 x - 7 y = 9$

Step 1.2

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

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Step 1.2.1

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

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

$5 x - 7 y = 9$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$5 x - 7 y = 9$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$5 x - 7 y = 9$

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

$5 x - 7 y = 9$

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

$5 x - 7 y = 9$

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

$5 x - 7 y = 9$

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

$5 x - 7 y = 9$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $5 x - 7 y = 9$ with $\frac{3}{2} + \frac{3 y}{2}$ .

$5 (\frac{3}{2} + \frac{3 y}{2}) - 7 y = 9$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 (\frac{3}{2} + \frac{3 y}{2}) - 7 y$ .

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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.

$5 (\frac{3}{2}) + 5 (\frac{3 y}{2}) - 7 y = 9$

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

Step 2.2.1.1.2

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

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Step 2.2.1.1.2.1

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

$\frac{5 \cdot 3}{2} + 5 (\frac{3 y}{2}) - 7 y = 9$

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

Step 2.2.1.1.2.2

Multiply $5$ by $3$ .

$\frac{15}{2} + 5 (\frac{3 y}{2}) - 7 y = 9$

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

$\frac{15}{2} + 5 (\frac{3 y}{2}) - 7 y = 9$

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

Step 2.2.1.1.3

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

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Step 2.2.1.1.3.1

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

$\frac{15}{2} + \frac{5 (3 y)}{2} - 7 y = 9$

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

Step 2.2.1.1.3.2

Multiply $3$ by $5$ .

$\frac{15}{2} + \frac{15 y}{2} - 7 y = 9$

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

$\frac{15}{2} + \frac{15 y}{2} - 7 y = 9$

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

$\frac{15}{2} + \frac{15 y}{2} - 7 y = 9$

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

Step 2.2.1.2

To write $- 7 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{15}{2} + \frac{15 y}{2} - 7 y \cdot \frac{2}{2} = 9$

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

Step 2.2.1.3

Combine $- 7 y$ and $\frac{2}{2}$ .

$\frac{15}{2} + \frac{15 y}{2} + \frac{- 7 y \cdot 2}{2} = 9$

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

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{15}{2} + \frac{15 y - 7 y \cdot 2}{2} = 9$

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

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{15 + 15 y - 7 y \cdot 2}{2} = 9$

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

Step 2.2.1.6

Multiply $2$ by $- 7$ .

$\frac{15 + 15 y - 14 y}{2} = 9$

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

Step 2.2.1.7

Subtract $14 y$ from $15 y$ .

$\frac{15 + y}{2} = 9$

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

$\frac{15 + y}{2} = 9$

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

$\frac{15 + y}{2} = 9$

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

$\frac{15 + y}{2} = 9$

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

Step 3

Solve for $y$ in $\frac{15 + y}{2} = 9$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{15 + y}{2}) = 2 \cdot 9$

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

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

Cancel the common factor of $2$ .

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Step 3.2.1.1.1

Cancel the common factor.

$2 (\frac{15 + y}{2}) = 2 \cdot 9$

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

Step 3.2.1.1.2

Rewrite the expression.

$15 + y = 2 \cdot 9$

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

$15 + y = 2 \cdot 9$

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

$15 + y = 2 \cdot 9$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $2$ by $9$ .

$15 + y = 18$

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

$15 + y = 18$

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

$15 + y = 18$

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

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 $15$ from both sides of the equation.

$y = 18 - 15$

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

Step 3.3.2

Subtract $15$ from $18$ .

$y = 3$

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

$y = 3$

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

$y = 3$

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

Step 4

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

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Step 4.1

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

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

$y = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Combine the numerators over the common denominator.

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

$y = 3$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $3$ by $3$ .

$x = \frac{3 + 9}{2}$

$y = 3$

Step 4.2.1.2.2

Add $3$ and $9$ .

$x = \frac{12}{2}$

$y = 3$

Step 4.2.1.2.3

Divide $12$ by $2$ .

$x = 6$

$y = 3$

$x = 6$

$y = 3$

$x = 6$

$y = 3$

$x = 6$

$y = 3$

$x = 6$

$y = 3$

Step 5

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

$(6, 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(6, 3)$

Equation Form:

$x = 6, y = 3$

Step 7