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 \(y=\frac{6}{5} x-5\).