Find the Equation Using Two Points (1/4,-1/2) , (3/4,4)

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

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.

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

Step 5

Finding the slope $m$ .

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

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

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

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{1}{4}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Multiply $4$ by $4$ .

$m = \frac{16 + 4 (\frac{1}{2})}{3 - 1}$

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

Multiply $- 1$ by $- 1$ .

$m = \frac{16 + 4 (1 (\frac{1}{2}))}{3 - 1}$

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

$m = \frac{16 + 4 (\frac{1}{2})}{3 - 1}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$m = \frac{16 + 2 (2) (\frac{1}{2})}{3 - 1}$

Cancel the common factor.

$m = \frac{16 + 2 \cdot (2 (\frac{1}{2}))}{3 - 1}$

Rewrite the expression.

$m = \frac{16 + 2}{3 - 1}$

Add $16$ and $2$ .

$m = \frac{18}{3 - 1}$

Simplify the expression.

Subtract $1$ from $3$ .

$m = \frac{18}{2}$

Divide $18$ by $2$ .

$m = 9$

Step 6

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

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

$y = m x + b$

Substitute the value of $m$ into the equation.

$y = (9) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (9) \cdot (\frac{1}{4}) + b$

Substitute the value of $y$ into the equation.

$- \frac{1}{2} = (9) \cdot (\frac{1}{4}) + b$

Find the value of $b$ .

Rewrite the equation as $9 \cdot \frac{1}{4} + b = - \frac{1}{2}$ .

$9 \cdot \frac{1}{4} + b = - \frac{1}{2}$

Combine $9$ and $\frac{1}{4}$ .

$\frac{9}{4} + b = - \frac{1}{2}$

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

Subtract $\frac{9}{4}$ from both sides of the equation.

$b = - \frac{1}{2} - \frac{9}{4}$

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

$b = - \frac{1}{2} \cdot \frac{2}{2} - \frac{9}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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

$b = - \frac{2}{2 \cdot 2} - \frac{9}{4}$

Multiply $2$ by $2$ .

$b = - \frac{2}{4} - \frac{9}{4}$

Combine the numerators over the common denominator.

$b = \frac{- 2 - 9}{4}$

Subtract $9$ from $- 2$ .

$b = \frac{- 11}{4}$

Move the negative in front of the fraction.

$b = - \frac{11}{4}$

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.

$y = 9 x - \frac{11}{4}$

Step 8

Find the Equation Using Two Points (1/4,-1/2) , (3/4,4)

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