Algebra Examples

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

Step 1

Find the slope of the line between $(- 1, \frac{3}{2})$ and $(4, - 1)$ 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.

$m = \frac{- 1 - (\frac{3}{2})}{4 - (- 1)}$

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

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

Step 1.4.1.2

Combine.

$m = \frac{2 (- 1 - \frac{3}{2})}{2 (4 - (- 1))}$

Step 1.4.2

Apply the distributive property.

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

Step 1.4.3

Cancel the common factor of $2$ .

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Step 1.4.3.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Step 1.4.3.2

Cancel the common factor.

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

Step 1.4.3.3

Rewrite the expression.

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

Step 1.4.4

Simplify the numerator.

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Step 1.4.4.1

Multiply $2$ by $- 1$ .

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

Step 1.4.4.2

Subtract $3$ from $- 2$ .

$m = \frac{- 5}{2 \cdot 4 + 2 (- (- 1))}$

Step 1.4.5

Simplify the denominator.

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Step 1.4.5.1

Multiply $2$ by $4$ .

$m = \frac{- 5}{8 + 2 (- (- 1))}$

Step 1.4.5.2

Multiply $2 (- (- 1))$ .

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Step 1.4.5.2.1

Multiply $- 1$ by $- 1$ .

$m = \frac{- 5}{8 + 2 \cdot 1}$

Step 1.4.5.2.2

Multiply $2$ by $1$ .

$m = \frac{- 5}{8 + 2}$

Step 1.4.5.3

Add $8$ and $2$ .

$m = \frac{- 5}{10}$

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 $- 5$ and $10$ .

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Step 1.4.6.1.1

Factor $5$ out of $- 5$ .

$m = \frac{5 (- 1)}{10}$

Step 1.4.6.1.2

Cancel the common factors.

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Step 1.4.6.1.2.1

Factor $5$ out of $10$ .

$m = \frac{5 \cdot - 1}{5 \cdot 2}$

Step 1.4.6.1.2.2

Cancel the common factor.

$m = \frac{5 \cdot - 1}{5 \cdot 2}$

Step 1.4.6.1.2.3

Rewrite the expression.

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

Step 1.4.6.2

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

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

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

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 $- \frac{1}{2} \cdot (x + 1)$ .

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Step 4.1.1.1

Rewrite.

$y - \frac{3}{2} = 0 + 0 - \frac{1}{2} \cdot (x + 1)$

Step 4.1.1.2

Simplify by adding zeros.

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

Step 4.1.1.3

Apply the distributive property.

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

Step 4.1.1.4

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

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

Step 4.1.1.5

Multiply $- 1$ by $1$ .

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

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 $\frac{3}{2}$ to both sides of the equation.

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

Step 4.1.2.2

Combine the numerators over the common denominator.

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

Step 4.1.2.3

Add $- 1$ and $3$ .

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

Step 4.1.2.4

Split the fraction $\frac{- x + 2}{2}$ into two fractions.

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

Step 4.1.2.5

Simplify each term.

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Step 4.1.2.5.1

Move the negative in front of the fraction.

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

Step 4.1.2.5.2

Divide $2$ by $2$ .

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

Step 4.2

Reorder terms.

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

Step 4.3

Remove parentheses.

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

Step 5