Algebra Examples

$(3, 6)$ and $(5, 5)$

Step 1

Use $y = m x + b$ to calculate the equation of the line, where $m$ represents the slope and $b$ represents the y-intercept.

To calculate the equation of the line, use the $y = m x + b$ format.

Step 2

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 3

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 4

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

$m = \frac{5 - (6)}{5 - (3)}$

Step 5

Finding the slope $m$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Multiply $- 1$ by $6$ .

$m = \frac{5 - 6}{5 - (3)}$

Step 5.1.2

Subtract $6$ from $5$ .

$m = \frac{- 1}{5 - (3)}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- 1$ by $3$ .

$m = \frac{- 1}{5 - 3}$

Step 5.2.2

Subtract $3$ from $5$ .

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

Step 5.3

Move the negative in front of the fraction.

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

Step 6

Find the value of $b$ using the formula for the equation of a line.

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Step 6.1

Use the formula for the equation of a line to find $b$ .

$y = m x + b$

Step 6.2

Substitute the value of $m$ into the equation.

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

Step 6.3

Substitute the value of $x$ into the equation.

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

Step 6.4

Substitute the value of $y$ into the equation.

$6 = (- \frac{1}{2}) \cdot (3) + b$

Step 6.5

Find the value of $b$ .

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Step 6.5.1

Rewrite the equation as $- \frac{1}{2} \cdot 3 + b = 6$ .

$- \frac{1}{2} \cdot 3 + b = 6$

Step 6.5.2

Simplify each term.

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Step 6.5.2.1

Multiply $- \frac{1}{2} \cdot 3$ .

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Step 6.5.2.1.1

Multiply $3$ by $- 1$ .

$- 3 (\frac{1}{2}) + b = 6$

Step 6.5.2.1.2

Combine $- 3$ and $\frac{1}{2}$ .

$\frac{- 3}{2} + b = 6$

Step 6.5.2.2

Move the negative in front of the fraction.

$- \frac{3}{2} + b = 6$

Step 6.5.3

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

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Step 6.5.3.1

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

$b = 6 + \frac{3}{2}$

Step 6.5.3.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$b = 6 \cdot \frac{2}{2} + \frac{3}{2}$

Step 6.5.3.3

Combine $6$ and $\frac{2}{2}$ .

$b = \frac{6 \cdot 2}{2} + \frac{3}{2}$

Step 6.5.3.4

Combine the numerators over the common denominator.

$b = \frac{6 \cdot 2 + 3}{2}$

Step 6.5.3.5

Simplify the numerator.

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Step 6.5.3.5.1

Multiply $6$ by $2$ .

$b = \frac{12 + 3}{2}$

Step 6.5.3.5.2

Add $12$ and $3$ .

$b = \frac{15}{2}$

Step 7

Now that the values of $m$ (slope) and $b$ (y-intercept) are known, substitute them into $y = m x + b$ to find the equation of the line.

$y = - \frac{1}{2} x + \frac{15}{2}$

Step 8