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Section 6.7 Additional Practice Related to Lines

Exercises Exercises

Questions vary.

1.

Complete the entries in Table 6.7.1 for the line with equation \(2x-5y=10\text{.}\) Also, state the \(x\) and \(y\) intercepts of the line.

\(x\) \(y\)
\(-10\)
\(-5\)
\(0\)
\(5\)
\(10\)
Table 6.7.1. \(2x-5y=10\)
Solution

The \(x\)-intercept is \((5,0)\) and the \(y\)-intercept is \((0,-2)\text{.}\)

\(x\) \(y\)
\(-10\) \(-6\)
\(-5\) \(-4\)
\(0\) \(-2\)
\(5\) \(0\)
\(10\) \(2\)
Table 6.7.2. \(2x-5y=10\)
2.

Complete the entries in Table 6.7.3 for the line with equation \(-x-7y=3\text{.}\) Also, state the \(x\) and \(y\) intercepts of the line.

\(x\) \(y\)
\(2\)
\(-3\)
\(-2\)
\(\frac{2}{7}\)
\(-\frac{5}{14}\)
Table 6.7.3. \(-x-7y=3\)
Solution

The \(x\)-intercept is \((-3,0)\) and the \(y\)-intercept is \(\left(0,-\frac{3}{7}\right)\text{.}\)

\(x\) \(y\)
\(2\) \(-\frac{5}{7}\)
\(-24\) \(-3\)
\(-2\) \(-\frac{1}{7}\)
\(-5\) \(\frac{2}{7}\)
\(-\frac{1}{2}\) \(-\frac{5}{14}\)
Table 6.7.4. \(-x-7y=3\)
3.

Determine the slope of the line that passes through the points \((2,-2)\) and \((-4,16)\text{.}\)

Solution

The slope is \(-3\text{.}\)

4.

Determine the slope of the line that passes through the points \((6,2)\) and \((-6,-8)\text{.}\)

Solution

The slope is \(\frac{5}{6}\text{.}\)

5.

Determine the slope of the line shown in Figure 6.7.5.

A graph of the line that passes through the points \((-3,5)\text{,}\) \((0,3)\text{,}\) and \((3,1)\text{.}\)
Figure 6.7.5. Determine the slope of the line
Solution

The slope is \(-\frac{2}{3}\text{.}\)

6.

A line with a slope of \(\frac{4}{5}\) passes through the point \((-4,-7)\text{.}\) What is the \(y\)-coordinate of the point on this line that has an \(x\)-coordinate of \(6\text{?}\)

Solution

The \(y\)-coordinate is \(1\text{.}\)

7.

A line with a slope of \(-3\) passes through the point \((4,-14)\text{.}\) What is the \(y\)-coordinate of the point that has an \(x\)-coordinate of \(-3\text{?}\)

Solution

The \(y\)-coordinate is \(7\text{.}\)

8.

A line with a slope of \(-\frac{1}{2}\) passes through the point \((410,27)\text{.}\) What is the \(x\)-coordinate of the point that a \(y\)-coordinate of \(24\text{?}\)

Solution

The \(x\)-coordinate is \(416\text{.}\)

Use the slope-intercept form of the equation of a line to determine the equation of each described line.

9.

Determine the equation of the line that has a slope of \(\frac{11}{8}\) and a \(y\)-intercept of \((0,-4)\text{.}\) Write the equation in slope-intercept form.

Solution

The equation is \(y=\frac{11}{8}x-4\)

10.

Determine the equation of the line that has a slope of \(-\frac{1}{3}\) and an \(x\)-intercept of \((6,0)\text{.}\) Write the equation in slope-intercept form.

Solution

The equation is \(y=-\frac{1}{3}x+2\text{.}\)

11.

Determine the equation of the line that passes through the points \((1,12)\) and \((-3,-8)\text{.}\) Write the equation in slope-intercept form.

Solution

The equation is \(y=5x+7\text{.}\)

12.

Determine the equation of the line shown in Figure 6.7.6. Write the equation in slope-intercept form.

A graph of the line that passes through the points \((-1,-3)\text{,}\) \((0,-1)\text{,}\) and \((1,1)\text{.}\)
Figure 6.7.6. Determine the slope of the line
Solution

The equation is \(y=\frac{5}{2}x-1\text{.}\)

Use the point-slope form of the equation of a line to determine the equation of the line that passes through each pair of points. State your final answer in slope-intercept form.

13.

\((9,-2)\) and \((7,-8)\)

Solution

The equation of the line is \(y=3x-29\text{.}\)

14.

\((-3,5)\) and \((5,1)\)

Solution

The equation of the line is \(y=-\frac{1}{2}x+\frac{7}{2}\text{.}\)

15.

\((-11,0)\) and \((-9,2)\)

Solution

The equation of the line is \(y=x+11\text{.}\)

16.

\((0,7)\) and \((-3,10)\)

Solution

The equation of the line is \(y=-x+7\text{.}\)

Questions vary.

17.

What are the equations of the vertical and horizontal lines that pass through the point \((9,2)\text{?}\)

Solution

The vertical line's equation is \(x=9\) and the horizontal line's equation is \(y=2\text{.}\)

18.

What is the slope of any line that is parallel to the line with equation \(6x-10y=7\text{?}\)

Solution

The slope is \(\frac{3}{5}\text{.}\)

19.

What is the slope of any line that is perpendicular to the line with equation \(y=-\frac{1}{8}x+11\text{?}\)

Solution

The slope is \(8\text{.}\)