Forms of linear equations: summary

1. What is the slope of the line?

$7 x+2 y=5$

Choose 1 answer:

A. $-\frac{7}{2}$

B. $\frac{7}{2}$

C. $-\frac{2}{7}$

D. $\frac{2}{7}$

2. What is the slope of the line?

$3(y-1)=2 x+2$

Choose 1 answer:

A. $\frac{5}{3}$

B. $\frac{1}{3}$

C. $\frac{2}{3}$

D. $\frac{4}{3}$

3. What is the slope of the line?

$8x-6y=1$

Choose 1 answer:

A. $\frac{1}{6}$

B. $\frac{4}{3}$

C. $\frac{3}{4}$

D. $\frac{1}{8}$

4. What is the slope of the line?

$y+1=3(x-4)$

Choose 1 answer:

A. $-\frac{4}{3}$

B. $-\frac{3}{4}$

C. $3$

D. $\frac{1}{3}$

1. A. $-\frac{7}{2}$

We can determine the slope of the graph by bringing the equation to slope-intercept form. So let’s solve the equation for $y$:

$\begin{aligned}7 x+2 y &=5 \\2 y &=5-7 x \\y &=\frac{5}{2}-\frac{7}{2} x\end{aligned}$

Now we have the equation in slope-intercept form: $y=m \cdot x+b$. In this form, the slope is simply the coefficient of $x$, meaning the value of $m$.

The slope is $-\frac{7}{2}$.

2. C. $\frac{2}{3}$

We can determine the slope of the graph by bringing the equation to slope-intercept form. So let’s solve the equation for $y$:

$\begin{array}{r} 3(y-1)=2 x+2 \\ 3 y-3=2 x+2 \\ 3 y=2 x+5 \\ y=\frac{2}{3} x+\frac{5}{3} \end{array}$

Now we have the equation in slope-intercept form: $y=m \cdot x+b$. In this form, the slope is simply the coefficient of $x$, meaning the value of $m$.

The slope is $\frac{2}{3}$

3. B. $\frac{4}{3}$

We can determine the slope of the graph by bringing the equation to slope-intercept form. So let’s solve the equation for $y$:

$\begin{aligned} &8 x-6 y=1 \\ &8 x-1=6 y \\ &\frac{8}{6} x-\frac{1}{6}=y \\ &\frac{4}{3} x-\frac{1}{6}=y \end{aligned}$

Now we have the equation in slope-intercept form: $y=m \cdot x+b$. In this form, the slope is simply the coefficient of $x$, meaning the value of $m$.

The slope is $\frac{4}{3}$

4. C. $3$

The equation is given in point-slope form, $y-y_{1}=m\left(x-x_{1}\right)$.

In this form, $m$ is the slope.

The slope is $3$.

1. $y+2=\frac{4}{5}(x-2)$.

The line passes through $(-3,-6)$ and $(2,-2)$.

We don't have the $y$-intercept so it's most comfortable to write an equation in point-slope form.

$\begin{aligned} \text { Slope } &=\frac{(-2)-(-6)}{2-(-3)} \\ &=\frac{4}{5} \end{aligned}$

Using the point $(2,-2)$, an equation that represents the line is $y+2=\frac{4}{5}(x-2)$.

2. $y=\frac{3}{2} x+3$.

The line passes through $(0,3)$ and $(2,6)$.

We have the $y$-intercept so it's most comfortable to find the slope-intercept form of the line.

$\begin{aligned} \text { Slope } &=\frac{6-3}{2-0} \\ &=\frac{3}{2} \end{aligned}$

An equation that represents the line is $y=\frac{3}{2} x+3$.

3. $y-4=\frac{7}{5}(x+2)$.

The line passes through $(-7,-3)$ and $(-2,4)$.

We don't have the $y$-intercept so it's most comfortable to write an equation in point-slope form.

$\begin{aligned} \text { Slope } &=\frac{4-(-3)}{(-2)-(-7)} \\ &=\frac{7}{5} \end{aligned}$

Using the point $(-2,4)$, an equation that represents the line is $y-4=\frac{7}{5}(x+2)$.

4.  $y=\frac{6}{5} x-5$

The line passes through $(0,-5)$ and $(5,1)$.

We have the $y$-intercept so it's most comfortable to find the slope-intercept form of the line.

$\begin{aligned} \text { Slope } &=\frac{1-(-5)}{5-0} \\ &=\frac{6}{5} \end{aligned}$

An equation that represents the line is $y=\frac{6}{5} x-5$.