Algebra Examples

$(- 6, \frac{5}{2})$ , $(4, 10)$

Step 1

Find the slope of the line between $(- 6, \frac{5}{2})$ and $(4, 10)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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Step 1.1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 1.2

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 1.3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{10 - (\frac{5}{2})}{4 - (- 6)}$

Step 1.4

Simplify.

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Step 1.4.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 1.4.1.1

Multiply $\frac{10 - \frac{5}{2}}{4 - (- 6)}$ by $\frac{2}{2}$ .

$m = \frac{2}{2} \cdot \frac{10 - \frac{5}{2}}{4 - (- 6)}$

Step 1.4.1.2

Combine.

$m = \frac{2 (10 - \frac{5}{2})}{2 (4 - (- 6))}$

Step 1.4.2

Apply the distributive property.

$m = \frac{2 \cdot 10 + 2 (- \frac{5}{2})}{2 \cdot 4 + 2 (- (- 6))}$

Step 1.4.3

Cancel the common factor of $2$ .

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Step 1.4.3.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$m = \frac{2 \cdot 10 + 2 (\frac{- 5}{2})}{2 \cdot 4 + 2 (- (- 6))}$

Step 1.4.3.2

Cancel the common factor.

$m = \frac{2 \cdot 10 + 2 (\frac{- 5}{2})}{2 \cdot 4 + 2 (- (- 6))}$

Step 1.4.3.3

Rewrite the expression.

$m = \frac{2 \cdot 10 - 5}{2 \cdot 4 + 2 (- (- 6))}$

Step 1.4.4

Simplify the numerator.

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Step 1.4.4.1

Multiply $2$ by $10$ .

$m = \frac{20 - 5}{2 \cdot 4 + 2 (- (- 6))}$

Step 1.4.4.2

Subtract $5$ from $20$ .

$m = \frac{15}{2 \cdot 4 + 2 (- (- 6))}$

Step 1.4.5

Simplify the denominator.

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Step 1.4.5.1

Multiply $2$ by $4$ .

$m = \frac{15}{8 + 2 (- (- 6))}$

Step 1.4.5.2

Multiply $2 (- (- 6))$ .

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Step 1.4.5.2.1

Multiply $- 1$ by $- 6$ .

$m = \frac{15}{8 + 2 \cdot 6}$

Step 1.4.5.2.2

Multiply $2$ by $6$ .

$m = \frac{15}{8 + 12}$

Step 1.4.5.3

Add $8$ and $12$ .

$m = \frac{15}{20}$

Step 1.4.6

Cancel the common factor of $15$ and $20$ .

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Step 1.4.6.1

Factor $5$ out of $15$ .

$m = \frac{5 (3)}{20}$

Step 1.4.6.2

Cancel the common factors.

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Step 1.4.6.2.1

Factor $5$ out of $20$ .

$m = \frac{5 \cdot 3}{5 \cdot 4}$

Step 1.4.6.2.2

Cancel the common factor.

$m = \frac{5 \cdot 3}{5 \cdot 4}$

Step 1.4.6.2.3

Rewrite the expression.

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

Step 2

Use the slope $\frac{3}{4}$ and a given point $(- 6, \frac{5}{2})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{5}{2}) = \frac{3}{4} \cdot (x - (- 6))$

Step 3

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

$y - \frac{5}{2} = \frac{3}{4} \cdot (x + 6)$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $\frac{3}{4} \cdot (x + 6)$ .

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Step 4.1.1

Rewrite.

$y - \frac{5}{2} = 0 + 0 + \frac{3}{4} \cdot (x + 6)$

Step 4.1.2

Simplify by adding zeros.

$y - \frac{5}{2} = \frac{3}{4} \cdot (x + 6)$

Step 4.1.3

Apply the distributive property.

$y - \frac{5}{2} = \frac{3}{4} x + \frac{3}{4} \cdot 6$

Step 4.1.4

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

$y - \frac{5}{2} = \frac{3 x}{4} + \frac{3}{4} \cdot 6$

Step 4.1.5

Cancel the common factor of $2$ .

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Step 4.1.5.1

Factor $2$ out of $4$ .

$y - \frac{5}{2} = \frac{3 x}{4} + \frac{3}{2 (2)} \cdot 6$

Step 4.1.5.2

Factor $2$ out of $6$ .

$y - \frac{5}{2} = \frac{3 x}{4} + \frac{3}{2 \cdot 2} \cdot (2 \cdot 3)$

Step 4.1.5.3

Cancel the common factor.

$y - \frac{5}{2} = \frac{3 x}{4} + \frac{3}{2 \cdot 2} \cdot (2 \cdot 3)$

Step 4.1.5.4

Rewrite the expression.

$y - \frac{5}{2} = \frac{3 x}{4} + \frac{3}{2} \cdot 3$

Step 4.1.6

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

$y - \frac{5}{2} = \frac{3 x}{4} + \frac{3 \cdot 3}{2}$

Step 4.1.7

Multiply $3$ by $3$ .

$y - \frac{5}{2} = \frac{3 x}{4} + \frac{9}{2}$

Step 4.2

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

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Step 4.2.1

Add $\frac{5}{2}$ to both sides of the equation.

$y = \frac{3 x}{4} + \frac{9}{2} + \frac{5}{2}$

Step 4.2.2

Combine the numerators over the common denominator.

$y = \frac{3 x}{4} + \frac{9 + 5}{2}$

Step 4.2.3

Add $9$ and $5$ .

$y = \frac{3 x}{4} + \frac{14}{2}$

Step 4.2.4

Divide $14$ by $2$ .

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

Step 5

Reorder terms.

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

Step 6

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

$y - \frac{5}{2} = \frac{3}{4} \cdot (x + 6)$

Step 7