Algebra Examples

$(1, 1)$ , $(8, - \frac{3}{4})$

Step 1

Find the slope of the line between $(1, 1)$ and $(8, - \frac{3}{4})$ 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.

$m = \frac{- \frac{3}{4} - (1)}{8 - (1)}$

Step 1.4

Simplify.

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Step 1.4.1

Multiply the numerator and denominator of the fraction by $4$ .

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Step 1.4.1.1

Multiply $\frac{- \frac{3}{4} - (1)}{8 - (1)}$ by $\frac{4}{4}$ .

$m = \frac{4}{4} \cdot \frac{- \frac{3}{4} - (1)}{8 - (1)}$

Step 1.4.1.2

Combine.

$m = \frac{4 (- \frac{3}{4} - (1))}{4 (8 - (1))}$

Step 1.4.2

Apply the distributive property.

$m = \frac{4 (- \frac{3}{4}) + 4 (- (1))}{4 \cdot 8 + 4 (- (1))}$

Step 1.4.3

Cancel the common factor of $4$ .

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Step 1.4.3.1

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

$m = \frac{4 (\frac{- 3}{4}) + 4 (- (1))}{4 \cdot 8 + 4 (- (1))}$

Step 1.4.3.2

Cancel the common factor.

$m = \frac{4 (\frac{- 3}{4}) + 4 (- (1))}{4 \cdot 8 + 4 (- (1))}$

Step 1.4.3.3

Rewrite the expression.

$m = \frac{- 3 + 4 (- (1))}{4 \cdot 8 + 4 (- (1))}$

Step 1.4.4

Simplify the numerator.

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Step 1.4.4.1

Multiply $4 (- (1))$ .

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Step 1.4.4.1.1

Multiply $- 1$ by $1$ .

$m = \frac{- 3 + 4 \cdot - 1}{4 \cdot 8 + 4 (- (1))}$

Step 1.4.4.1.2

Multiply $4$ by $- 1$ .

$m = \frac{- 3 - 4}{4 \cdot 8 + 4 (- (1))}$

Step 1.4.4.2

Subtract $4$ from $- 3$ .

$m = \frac{- 7}{4 \cdot 8 + 4 (- (1))}$

Step 1.4.5

Simplify the denominator.

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Step 1.4.5.1

Multiply $4$ by $8$ .

$m = \frac{- 7}{32 + 4 (- (1))}$

Step 1.4.5.2

Multiply $4 (- (1))$ .

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Step 1.4.5.2.1

Multiply $- 1$ by $1$ .

$m = \frac{- 7}{32 + 4 \cdot - 1}$

Step 1.4.5.2.2

Multiply $4$ by $- 1$ .

$m = \frac{- 7}{32 - 4}$

Step 1.4.5.3

Subtract $4$ from $32$ .

$m = \frac{- 7}{28}$

Step 1.4.6

Reduce the expression by cancelling the common factors.

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Step 1.4.6.1

Cancel the common factor of $- 7$ and $28$ .

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Step 1.4.6.1.1

Factor $7$ out of $- 7$ .

$m = \frac{7 (- 1)}{28}$

Step 1.4.6.1.2

Cancel the common factors.

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Step 1.4.6.1.2.1

Factor $7$ out of $28$ .

$m = \frac{7 \cdot - 1}{7 \cdot 4}$

Step 1.4.6.1.2.2

Cancel the common factor.

$m = \frac{7 \cdot - 1}{7 \cdot 4}$

Step 1.4.6.1.2.3

Rewrite the expression.

$m = \frac{- 1}{4}$

Step 1.4.6.2

Move the negative in front of the fraction.

$m = - \frac{1}{4}$

Step 2

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

$y - (1) = - \frac{1}{4} \cdot (x - (1))$

Step 3

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

$y - 1 = - \frac{1}{4} \cdot (x - 1)$

Step 4

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

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Step 4.1

Solve for $y$ .

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Step 4.1.1

Simplify $- \frac{1}{4} \cdot (x - 1)$ .

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Step 4.1.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{1}{4} \cdot (x - 1)$

Step 4.1.1.2

Simplify by adding zeros.

$y - 1 = - \frac{1}{4} \cdot (x - 1)$

Step 4.1.1.3

Apply the distributive property.

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

Step 4.1.1.4

Combine $x$ and $\frac{1}{4}$ .

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

Step 4.1.1.5

Multiply $- \frac{1}{4} \cdot - 1$ .

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Step 4.1.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.1.5.2

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

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

Step 4.1.2

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

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Step 4.1.2.1

Add $1$ to both sides of the equation.

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

Step 4.1.2.2

Write $1$ as a fraction with a common denominator.

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

Step 4.1.2.3

Combine the numerators over the common denominator.

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

Step 4.1.2.4

Add $1$ and $4$ .

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

Step 4.2

Reorder terms.

$y = - (\frac{1}{4} x) + \frac{5}{4}$

Step 4.3

Remove parentheses.

$y = - \frac{1}{4} x + \frac{5}{4}$

Step 5