Algebra Examples

$3 x + 4 y = 7$ , $(- \frac{2}{3}, \frac{7}{8})$

Step 1

Rewrite in slope-intercept form.

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Step 1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 1.2

Subtract $3 x$ from both sides of the equation.

$4 y = 7 - 3 x$

Step 1.3

Divide each term in $4 y = 7 - 3 x$ by $4$ and simplify.

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Step 1.3.1

Divide each term in $4 y = 7 - 3 x$ by $4$ .

$\frac{4 y}{4} = \frac{7}{4} + \frac{- 3 x}{4}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $4$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{7}{4} + \frac{- 3 x}{4}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{7}{4} + \frac{- 3 x}{4}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Move the negative in front of the fraction.

$y = \frac{7}{4} - \frac{3 x}{4}$

Step 1.4

Write in $y = m x + b$ form.

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Step 1.4.1

Reorder $\frac{7}{4}$ and $- \frac{3 x}{4}$ .

$y = - \frac{3 x}{4} + \frac{7}{4}$

Step 1.4.2

Reorder terms.

$y = - (\frac{3}{4} x) + \frac{7}{4}$

Step 1.4.3

Remove parentheses.

$y = - \frac{3}{4} x + \frac{7}{4}$

Step 2

Using the slope-intercept form, the slope is $- \frac{3}{4}$ .

$m = - \frac{3}{4}$

Step 3

To find an equation that is parallel, the slopes must be equal. Find the parallel line using the point-slope formula.

Step 4

Use the slope $- \frac{3}{4}$ and a given point $(- \frac{2}{3}, \frac{7}{8})$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{7}{8}) = - \frac{3}{4} \cdot (x - (- \frac{2}{3}))$

Step 5

Simplify the equation and keep it in point-slope form.

$y - \frac{7}{8} = - \frac{3}{4} \cdot (x + \frac{2}{3})$

Step 6

Solve for $y$ .

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Step 6.1

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

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Step 6.1.1

Rewrite.

$y - \frac{7}{8} = 0 + 0 - \frac{3}{4} \cdot (x + \frac{2}{3})$

Step 6.1.2

Simplify terms.

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Step 6.1.2.1

Apply the distributive property.

$y - \frac{7}{8} = - \frac{3}{4} x - \frac{3}{4} \cdot \frac{2}{3}$

Step 6.1.2.2

Combine $x$ and $\frac{3}{4}$ .

$y - \frac{7}{8} = - \frac{x \cdot 3}{4} - \frac{3}{4} \cdot \frac{2}{3}$

Step 6.1.2.3

Cancel the common factor of $3$ .

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Step 6.1.2.3.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$y - \frac{7}{8} = - \frac{x \cdot 3}{4} + \frac{- 3}{4} \cdot \frac{2}{3}$

Step 6.1.2.3.2

Factor $3$ out of $- 3$ .

$y - \frac{7}{8} = - \frac{x \cdot 3}{4} + \frac{3 (- 1)}{4} \cdot \frac{2}{3}$

Step 6.1.2.3.3

Cancel the common factor.

$y - \frac{7}{8} = - \frac{x \cdot 3}{4} + \frac{3 \cdot - 1}{4} \cdot \frac{2}{3}$

Step 6.1.2.3.4

Rewrite the expression.

$y - \frac{7}{8} = - \frac{x \cdot 3}{4} + \frac{- 1}{4} \cdot 2$

Step 6.1.2.4

Cancel the common factor of $2$ .

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Step 6.1.2.4.1

Factor $2$ out of $4$ .

$y - \frac{7}{8} = - \frac{x \cdot 3}{4} + \frac{- 1}{2 (2)} \cdot 2$

Step 6.1.2.4.2

Cancel the common factor.

$y - \frac{7}{8} = - \frac{x \cdot 3}{4} + \frac{- 1}{2 \cdot 2} \cdot 2$

Step 6.1.2.4.3

Rewrite the expression.

$y - \frac{7}{8} = - \frac{x \cdot 3}{4} + \frac{- 1}{2}$

Step 6.1.3

Simplify each term.

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Step 6.1.3.1

Move $3$ to the left of $x$ .

$y - \frac{7}{8} = - \frac{3 \cdot x}{4} + \frac{- 1}{2}$

Step 6.1.3.2

Move the negative in front of the fraction.

$y - \frac{7}{8} = - \frac{3 x}{4} - \frac{1}{2}$

Step 6.2

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

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Step 6.2.1

Add $\frac{7}{8}$ to both sides of the equation.

$y = - \frac{3 x}{4} - \frac{1}{2} + \frac{7}{8}$

Step 6.2.2

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

$y = - \frac{3 x}{4} - \frac{1}{2} \cdot \frac{4}{4} + \frac{7}{8}$

Step 6.2.3

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

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Step 6.2.3.1

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

$y = - \frac{3 x}{4} - \frac{4}{2 \cdot 4} + \frac{7}{8}$

Step 6.2.3.2

Multiply $2$ by $4$ .

$y = - \frac{3 x}{4} - \frac{4}{8} + \frac{7}{8}$

Step 6.2.4

Combine the numerators over the common denominator.

$y = - \frac{3 x}{4} + \frac{- 4 + 7}{8}$

Step 6.2.5

Add $- 4$ and $7$ .

$y = - \frac{3 x}{4} + \frac{3}{8}$

Step 6.3

Write in $y = m x + b$ form.

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Step 6.3.1

Reorder terms.

$y = - (\frac{3}{4} x) + \frac{3}{8}$

Step 6.3.2

Remove parentheses.

$y = - \frac{3}{4} x + \frac{3}{8}$

Step 7