Precalculus Examples

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$ , $2 x - \frac{1}{2} y = - \frac{1}{2}$

Step 1

Solve for $x$ in $2 x - \frac{1}{2} y = - \frac{1}{2}$ .

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Step 1.1

Combine $y$ and $\frac{1}{2}$ .

$2 x - \frac{y}{2} = - \frac{1}{2}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

Step 1.2

Add $\frac{y}{2}$ to both sides of the equation.

$2 x = - \frac{1}{2} + \frac{y}{2}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

Step 1.3

Divide each term in $2 x = - \frac{1}{2} + \frac{y}{2}$ by $2$ and simplify.

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Step 1.3.1

Divide each term in $2 x = - \frac{1}{2} + \frac{y}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{- \frac{1}{2}}{2} + \frac{\frac{y}{2}}{2}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $2$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- \frac{1}{2}}{2} + \frac{\frac{y}{2}}{2}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{1}{2}}{2} + \frac{\frac{y}{2}}{2}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

$x = \frac{- \frac{1}{2}}{2} + \frac{\frac{y}{2}}{2}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

$x = \frac{- \frac{1}{2}}{2} + \frac{\frac{y}{2}}{2}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{1}{2} \cdot \frac{1}{2} + \frac{\frac{y}{2}}{2}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

Step 1.3.3.1.2

Multiply $- \frac{1}{2} \cdot \frac{1}{2}$ .

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Step 1.3.3.1.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$x = - \frac{1}{2 \cdot 2} + \frac{\frac{y}{2}}{2}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

Step 1.3.3.1.2.2

Multiply $2$ by $2$ .

$x = - \frac{1}{4} + \frac{\frac{y}{2}}{2}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{\frac{y}{2}}{2}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

Step 1.3.3.1.3

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{1}{4} + \frac{y}{2} \cdot \frac{1}{2}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

Step 1.3.3.1.4

Multiply $\frac{y}{2} \cdot \frac{1}{2}$ .

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Step 1.3.3.1.4.1

Multiply $\frac{y}{2}$ by $\frac{1}{2}$ .

$x = - \frac{1}{4} + \frac{y}{2 \cdot 2}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

Step 1.3.3.1.4.2

Multiply $2$ by $2$ .

$x = - \frac{1}{4} + \frac{y}{4}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

$\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$

Step 2

Replace all occurrences of $x$ with $- \frac{1}{4} + \frac{y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $\frac{3}{2} x - \frac{1}{3} y = \frac{1}{2}$ with $- \frac{1}{4} + \frac{y}{4}$ .

$\frac{3}{2} \cdot (- \frac{1}{4} + \frac{y}{4}) - \frac{1}{3} y = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{3}{2} \cdot (- \frac{1}{4} + \frac{y}{4}) - \frac{1}{3} y$ .

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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.

$\frac{3}{2} \cdot (- \frac{1}{4}) + \frac{3}{2} \cdot \frac{y}{4} - \frac{1}{3} y = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.1.2

Multiply $\frac{3}{2} (- \frac{1}{4})$ .

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Step 2.2.1.1.2.1

Multiply $\frac{3}{2}$ by $\frac{1}{4}$ .

$- \frac{3}{2 \cdot 4} + \frac{3}{2} \cdot \frac{y}{4} - \frac{1}{3} y = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.1.2.2

Multiply $2$ by $4$ .

$- \frac{3}{8} + \frac{3}{2} \cdot \frac{y}{4} - \frac{1}{3} y = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

$- \frac{3}{8} + \frac{3}{2} \cdot \frac{y}{4} - \frac{1}{3} y = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.1.3

Multiply $\frac{3}{2} \cdot \frac{y}{4}$ .

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Step 2.2.1.1.3.1

Multiply $\frac{3}{2}$ by $\frac{y}{4}$ .

$- \frac{3}{8} + \frac{3 y}{2 \cdot 4} - \frac{1}{3} y = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.1.3.2

Multiply $2$ by $4$ .

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

$x = - \frac{1}{4} + \frac{y}{4}$

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

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.1.4

Combine $y$ and $\frac{1}{3}$ .

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

$x = - \frac{1}{4} + \frac{y}{4}$

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

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.2

To write $\frac{3 y}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{3}{8} + \frac{3 y}{8} \cdot \frac{3}{3} - \frac{y}{3} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.3

To write $- \frac{y}{3}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{3}{8} + \frac{3 y}{8} \cdot \frac{3}{3} - \frac{y}{3} \cdot \frac{8}{8} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.4

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.4.1

Multiply $\frac{3 y}{8}$ by $\frac{3}{3}$ .

$- \frac{3}{8} + \frac{3 y \cdot 3}{8 \cdot 3} - \frac{y}{3} \cdot \frac{8}{8} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.4.2

Multiply $8$ by $3$ .

$- \frac{3}{8} + \frac{3 y \cdot 3}{24} - \frac{y}{3} \cdot \frac{8}{8} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.4.3

Multiply $\frac{y}{3}$ by $\frac{8}{8}$ .

$- \frac{3}{8} + \frac{3 y \cdot 3}{24} - \frac{y \cdot 8}{3 \cdot 8} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.4.4

Multiply $3$ by $8$ .

$- \frac{3}{8} + \frac{3 y \cdot 3}{24} - \frac{y \cdot 8}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

$- \frac{3}{8} + \frac{3 y \cdot 3}{24} - \frac{y \cdot 8}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$- \frac{3}{8} + \frac{3 y \cdot 3 - y \cdot 8}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.6

Simplify each term.

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Step 2.2.1.6.1

Simplify the numerator.

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Step 2.2.1.6.1.1

Factor $y$ out of $3 y \cdot 3 - y \cdot 8$ .

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Step 2.2.1.6.1.1.1

Factor $y$ out of $3 y \cdot 3$ .

$- \frac{3}{8} + \frac{y (3 \cdot 3) - y \cdot 8}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.6.1.1.2

Factor $y$ out of $- y \cdot 8$ .

$- \frac{3}{8} + \frac{y (3 \cdot 3) + y (- 1 \cdot 8)}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.6.1.1.3

Factor $y$ out of $y (3 \cdot 3) + y (- 1 \cdot 8)$ .

$- \frac{3}{8} + \frac{y (3 \cdot 3 - 1 \cdot 8)}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

$- \frac{3}{8} + \frac{y (3 \cdot 3 - 1 \cdot 8)}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.6.1.2

Multiply $3$ by $3$ .

$- \frac{3}{8} + \frac{y (9 - 1 \cdot 8)}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.6.1.3

Multiply $- 1$ by $8$ .

$- \frac{3}{8} + \frac{y (9 - 8)}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.6.1.4

Subtract $8$ from $9$ .

$- \frac{3}{8} + \frac{y \cdot 1}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

$- \frac{3}{8} + \frac{y \cdot 1}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 2.2.1.6.2

Multiply $y$ by $1$ .

$- \frac{3}{8} + \frac{y}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

$- \frac{3}{8} + \frac{y}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

$- \frac{3}{8} + \frac{y}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

$- \frac{3}{8} + \frac{y}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

$- \frac{3}{8} + \frac{y}{24} = \frac{1}{2}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 3

Solve for $y$ in $- \frac{3}{8} + \frac{y}{24} = \frac{1}{2}$ .

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Step 3.1

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

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Step 3.1.1

Add $\frac{3}{8}$ to both sides of the equation.

$\frac{y}{24} = \frac{1}{2} + \frac{3}{8}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 3.1.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{y}{24} = \frac{1}{2} \cdot \frac{4}{4} + \frac{3}{8}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 3.1.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1.3.1

Multiply $\frac{1}{2}$ by $\frac{4}{4}$ .

$\frac{y}{24} = \frac{4}{2 \cdot 4} + \frac{3}{8}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 3.1.3.2

Multiply $2$ by $4$ .

$\frac{y}{24} = \frac{4}{8} + \frac{3}{8}$

$x = - \frac{1}{4} + \frac{y}{4}$

$\frac{y}{24} = \frac{4}{8} + \frac{3}{8}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{y}{24} = \frac{4 + 3}{8}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 3.1.5

Add $4$ and $3$ .

$\frac{y}{24} = \frac{7}{8}$

$x = - \frac{1}{4} + \frac{y}{4}$

$\frac{y}{24} = \frac{7}{8}$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 3.2

Multiply both sides of the equation by $24$ .

$24 (\frac{y}{24}) = 24 (\frac{7}{8})$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $24$ .

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Step 3.3.1.1.1

Cancel the common factor.

$24 (\frac{y}{24}) = 24 (\frac{7}{8})$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 3.3.1.1.2

Rewrite the expression.

$y = 24 (\frac{7}{8})$

$x = - \frac{1}{4} + \frac{y}{4}$

$y = 24 (\frac{7}{8})$

$x = - \frac{1}{4} + \frac{y}{4}$

$y = 24 (\frac{7}{8})$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $24 (\frac{7}{8})$ .

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Step 3.3.2.1.1

Cancel the common factor of $8$ .

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Step 3.3.2.1.1.1

Factor $8$ out of $24$ .

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

$x = - \frac{1}{4} + \frac{y}{4}$

Step 3.3.2.1.1.2

Cancel the common factor.

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

$x = - \frac{1}{4} + \frac{y}{4}$

Step 3.3.2.1.1.3

Rewrite the expression.

$y = 3 \cdot 7$

$x = - \frac{1}{4} + \frac{y}{4}$

$y = 3 \cdot 7$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 3.3.2.1.2

Multiply $3$ by $7$ .

$y = 21$

$x = - \frac{1}{4} + \frac{y}{4}$

$y = 21$

$x = - \frac{1}{4} + \frac{y}{4}$

$y = 21$

$x = - \frac{1}{4} + \frac{y}{4}$

$y = 21$

$x = - \frac{1}{4} + \frac{y}{4}$

$y = 21$

$x = - \frac{1}{4} + \frac{y}{4}$

Step 4

Replace all occurrences of $y$ with $21$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{1}{4} + \frac{y}{4}$ with $21$ .

$x = - \frac{1}{4} + \frac{21}{4}$

$y = 21$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{1}{4} + \frac{21}{4}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 1 + 21}{4}$

$y = 21$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Add $- 1$ and $21$ .

$x = \frac{20}{4}$

$y = 21$

Step 4.2.1.2.2

Divide $20$ by $4$ .

$x = 5$

$y = 21$

$x = 5$

$y = 21$

$x = 5$

$y = 21$

$x = 5$

$y = 21$

$x = 5$

$y = 21$

Step 5

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

$(5, 21)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(5, 21)$

Equation Form:

$x = 5, y = 21$

Step 7