Algebra Examples

$2 x - 3 y = 13$ $y = \frac{1}{2} x - \frac{7}{2}$

Step 1

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

$y = \frac{x}{2} - \frac{7}{2}$

$2 x - 3 y = 13$

Step 2

Replace all occurrences of $y$ with $\frac{x}{2} - \frac{7}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $2 x - 3 y = 13$ with $\frac{x}{2} - \frac{7}{2}$ .

$2 x - 3 (\frac{x}{2} - \frac{7}{2}) = 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 x - 3 (\frac{x}{2} - \frac{7}{2})$ .

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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.

$2 x - 3 \frac{x}{2} - 3 (- \frac{7}{2}) = 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 2.2.1.1.2

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

$2 x + \frac{- 3 x}{2} - 3 (- \frac{7}{2}) = 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 2.2.1.1.3

Multiply $- 3 (- \frac{7}{2})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $- 3$ .

$2 x + \frac{- 3 x}{2} + 3 (\frac{7}{2}) = 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 2.2.1.1.3.2

Combine $3$ and $\frac{7}{2}$ .

$2 x + \frac{- 3 x}{2} + \frac{3 \cdot 7}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 2.2.1.1.3.3

Multiply $3$ by $7$ .

$2 x + \frac{- 3 x}{2} + \frac{21}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

$2 x + \frac{- 3 x}{2} + \frac{21}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 2.2.1.1.4

Move the negative in front of the fraction.

$2 x - \frac{3 x}{2} + \frac{21}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

$2 x - \frac{3 x}{2} + \frac{21}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 2.2.1.2

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

$2 x \cdot \frac{2}{2} - \frac{3 x}{2} + \frac{21}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 2.2.1.3

Combine $2 x$ and $\frac{2}{2}$ .

$\frac{2 x \cdot 2}{2} - \frac{3 x}{2} + \frac{21}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{2 x \cdot 2 - 3 x}{2} + \frac{21}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{2 x \cdot 2 - 3 x + 21}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 2.2.1.6

Multiply $2$ by $2$ .

$\frac{4 x - 3 x + 21}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 2.2.1.7

Subtract $3 x$ from $4 x$ .

$\frac{x + 21}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

$\frac{x + 21}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

$\frac{x + 21}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

$\frac{x + 21}{2} = 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 3

Solve for $x$ in $\frac{x + 21}{2} = 13$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{x + 21}{2}) = 2 \cdot 13$

$y = \frac{x}{2} - \frac{7}{2}$

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.

$2 (\frac{x + 21}{2}) = 2 \cdot 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 3.2.1.1.2

Rewrite the expression.

$x + 21 = 2 \cdot 13$

$y = \frac{x}{2} - \frac{7}{2}$

$x + 21 = 2 \cdot 13$

$y = \frac{x}{2} - \frac{7}{2}$

$x + 21 = 2 \cdot 13$

$y = \frac{x}{2} - \frac{7}{2}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $2$ by $13$ .

$x + 21 = 26$

$y = \frac{x}{2} - \frac{7}{2}$

$x + 21 = 26$

$y = \frac{x}{2} - \frac{7}{2}$

$x + 21 = 26$

$y = \frac{x}{2} - \frac{7}{2}$

Step 3.3

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

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Step 3.3.1

Subtract $21$ from both sides of the equation.

$x = 26 - 21$

$y = \frac{x}{2} - \frac{7}{2}$

Step 3.3.2

Subtract $21$ from $26$ .

$x = 5$

$y = \frac{x}{2} - \frac{7}{2}$

$x = 5$

$y = \frac{x}{2} - \frac{7}{2}$

$x = 5$

$y = \frac{x}{2} - \frac{7}{2}$

Step 4

Replace all occurrences of $x$ with $5$ in each equation.

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Step 4.1

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

$y = \frac{5}{2} - \frac{7}{2}$

$x = 5$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{5}{2} - \frac{7}{2}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$y = \frac{5 - 7}{2}$

$x = 5$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Subtract $7$ from $5$ .

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

$x = 5$

Step 4.2.1.2.2

Divide $- 2$ by $2$ .

$y = - 1$

$x = 5$

$y = - 1$

$x = 5$

$y = - 1$

$x = 5$

$y = - 1$

$x = 5$

$y = - 1$

$x = 5$

Step 5

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

$(5, - 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(5, - 1)$

Equation Form:

$x = 5, y = - 1$

Step 7