4.6E: Exercises

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Practice Makes Perfect

Find an Equation of the Line Given the Slope and $$y$$-Intercept

In the following exercises, find the equation of a line with given slope and $$y$$-intercept. Write the equation in slope–intercept form.

Exercise $$\PageIndex{1}$$

slope $$3$$ and $$y$$-intercept $$(0,5)$$

Exercise $$\PageIndex{2}$$

slope $$4$$ and $$y$$-intercept $$(0,1)$$

$$y=4x+1$$

Exercise $$\PageIndex{3}$$

slope $$6$$ and $$y$$-intercept $$(0,−4)$$

Exercise $$\PageIndex{4}$$

slope $$8$$ and $$y$$-intercept $$(0,−6)$$

$$y=8x−6$$

Exercise $$\PageIndex{5}$$

slope $$−1$$ and $$y$$-intercept $$(0,3)$$

Exercise $$\PageIndex{6}$$

slope $$−1$$ and $$y$$-intercept $$(0,7)$$

$$y=−x+7$$

Exercise $$\PageIndex{7}$$

slope $$−2$$ and $$y$$-intercept $$(0,−3)$$

Exercise $$\PageIndex{8}$$

slope $$−3$$ and $$y$$-intercept $$(0,−1)$$

$$y=−3x−1$$

Exercise $$\PageIndex{9}$$

slope $$\frac{3}{5}$$ and $$y$$-intercept $$(0,-1)$$

Exercise $$\PageIndex{10}$$

slope $$\frac{1}{5}$$ and $$y$$-intercept $$(0,-5)$$

$$y=\frac{1}{5} x-5$$

Exercise $$\PageIndex{11}$$

slope $$-\frac{3}{4}$$ and $$y$$-intercept $$(0,-2)$$

Exercise $$\PageIndex{12}$$

slope $$-\frac{2}{3}$$ and $$y$$-intercept $$(0,-3)$$

$$y=-\frac{2}{3} x-3$$

Exercise $$\PageIndex{13}$$

slope $$0$$ and $$y$$-intercept $$(0,-1)$$

Exercise $$\PageIndex{14}$$

slope $$0$$ and $$y$$-intercept $$(0,2)$$

$$y=2$$

Exercise $$\PageIndex{15}$$

slope $$-3$$ and $$y$$-intercept $$(0,0)$$

Exercise $$\PageIndex{16}$$

slope $$-4$$ and $$y$$-intercept $$(0,0)$$

$$y=−4x$$

In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.

Exercise $$\PageIndex{18}$$

$$y=−2x+4$$

Exercise $$\PageIndex{20}$$

$$y=\frac{3}{4} x+2$$

Exercise $$\PageIndex{22}$$

$$y=-\frac{3}{2} x-1$$

Exercise $$\PageIndex{24}$$

$$y=6$$

Find an Equation of the Line Given the Slope and a Point

In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.

Exercise $$\PageIndex{25}$$

$$m=\frac{5}{8},$$ point $$(8,3)$$

Exercise $$\PageIndex{26}$$

$$m=\frac{3}{8},$$ point $$(8,2)$$

$$y=\frac{3}{8} x-1$$

Exercise $$\PageIndex{27}$$

$$m=\frac{1}{6},$$ point $$(6,1)$$

Exercise $$\PageIndex{28}$$

$$m=\frac{5}{6},$$ point $$(6,7)$$

$$y=\frac{5}{6} x+2$$

Exercise $$\PageIndex{29}$$

$$m=-\frac{3}{4},$$ point $$(8,-5)$$

Exercise $$\PageIndex{30}$$

$$m=-\frac{3}{5},$$ point $$(10,-5)$$

$$y=-\frac{3}{5} x+1$$

Exercise $$\PageIndex{31}$$

$$m=-\frac{1}{4},$$ point $$(-12,-6)$$

Exercise $$\PageIndex{32}$$

$$m=-\frac{1}{3},$$ point $$(-9,-8)$$

$$y=-\frac{1}{3} x-11$$

Exercise $$\PageIndex{33}$$

Horizontal line containing $$(−2,5)$$

Exercise $$\PageIndex{34}$$

Horizontal line containing $$(−1,4)$$

$$y=4$$

Exercise $$\PageIndex{35}$$

Horizontal line containing $$(−2,−3)$$

Exercise $$\PageIndex{36}$$

Horizontal line containing $$(−1,−7)$$

$$y=−7$$

Exercise $$\PageIndex{37}$$

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

Exercise $$\PageIndex{38}$$

$$m=-\frac{5}{2},$$ point $$(-8,-2)$$

$$y=-\frac{5}{2} x-22$$

Exercise $$\PageIndex{39}$$

$$m=-7,$$ point $$(-1,-3)$$

Exercise $$\PageIndex{40}$$

$$m=-4,$$ point $$(-2,-3)$$

$$y=-4 x-11$$

Exercise $$\PageIndex{41}$$

Horizontal line containing $$(2,-3)$$

Exercise $$\PageIndex{42}$$

Horizontal line containing $$(4,-8)$$

$$y=−8$$

Find an Equation of the Line Given Two Points

In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.

Exercise $$\PageIndex{43}$$

$$(2,6)$$ and $$(5,3)$$

Exercise $$\PageIndex{44}$$

$$(3,1)$$ and $$(2,5)$$

$$y=−4x+13$$

Exercise $$\PageIndex{45}$$

$$(4,3)$$ and $$(8,1)$$

Exercise $$\PageIndex{46}$$

$$(2,7)$$ and $$(3,8)$$

$$y=x+5$$

Exercise $$\PageIndex{47}$$

$$(−3,−4)$$ and $$(5,−2)$$

Exercise $$\PageIndex{48}$$

$$(−5,−3)$$ and $$(4,−6)$$

$$y=-\frac{1}{3} x-\frac{14}{3}$$

Exercise $$\PageIndex{49}$$

$$(−1,3)$$ and $$(−6,−7)$$

Exercise $$\PageIndex{50}$$

$$(−2,8)$$ and $$(−4,−6)$$

$$y=7x+22$$

Exercise $$\PageIndex{51}$$

$$(6,−4)$$ and $$(−2,5)$$

Exercise $$\PageIndex{52}$$

$$(3,−2)$$ and $$(−4,4)$$

$$y=-\frac{6}{7} x+\frac{4}{7}$$

Exercise $$\PageIndex{53}$$

$$(0,4)$$ and $$(2,−3)$$

Exercise $$\PageIndex{54}$$

$$(0,−2)$$ and $$(−5,−3)$$

$$y=\frac{1}{5} x-2$$

Exercise $$\PageIndex{55}$$

$$(7,2)$$ and $$(7,−2)$$

Exercise $$\PageIndex{56}$$

$$(4,2)$$ and $$(4,−3)$$

$$x=4$$

Exercise $$\PageIndex{57}$$

$$(−7,−1)$$ and $$(−7,−4)$$

Exercise $$\PageIndex{58}$$

$$(−2,1)$$ and $$(−2,−4)$$

$$x=−2$$

Exercise $$\PageIndex{59}$$

$$(6,1)$$ and $$(0,1)$$

Exercise $$\PageIndex{60}$$

$$(6,2)$$ and $$(−3,2)$$

$$y=2$$

Exercise $$\PageIndex{61}$$

$$(3,−4)$$ and $$(5,−4)$$

Exercise $$\PageIndex{62}$$

$$(−6,−3)$$ and $$(−1,−3)$$

$$y=−3$$

Exercise $$\PageIndex{63}$$

$$(4,3)$$ and $$(8,0)$$

Exercise $$\PageIndex{64}$$

$$(0,0)$$ and $$(1,4)$$

$$y=4x$$

Exercise $$\PageIndex{65}$$

$$(−2,−3)$$ and $$(−5,−6)$$

Exercise $$\PageIndex{66}$$

$$(−3,0)$$ and $$(−7,−2)$$

$$y=\frac{1}{2} x+\frac{3}{2}$$

Exercise $$\PageIndex{67}$$

$$(8,−1)$$ and $$(8,−5)$$

Exercise $$\PageIndex{68}$$

$$(3,5)$$ and $$(−7,5)$$

$$y=5$$

Find an Equation of a Line Parallel to a Given Line

In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.

Exercise $$\PageIndex{69}$$

line $$y=4 x+2,$$ point $$(1,2)$$

Exercise $$\PageIndex{70}$$

line $$y=3 x+4,$$ point $$(2,5)$$

$$y=3 x-1$$

Exercise $$\PageIndex{71}$$

line $$y=-2 x-3,$$ point $$(-1,3)$$

Exercise $$\PageIndex{72}$$

line $$y=-3x-1,$$ point $$(2,-3)$$

$$y=−3x+3$$

Exercise $$\PageIndex{73}$$

line $$3 x-y=4,$$ point $$(3,1)$$

Exercise $$\PageIndex{74}$$

line $$2 x-y=6,$$ point $$(3,0)$$

$$y=2x−6$$

Exercise $$\PageIndex{75}$$

line $$4 x+3 y=6,$$ point $$(0,-3)$$

Exercise $$\PageIndex{76}$$

line $$2x+3y=6,$$ point $$(0,5)$$

$$y=-\frac{2}{3} x+5$$

Exercise $$\PageIndex{77}$$

line $$x=-3,$$ point $$(-2,-1)$$

Exercise $$\PageIndex{78}$$

line $$x=-4,$$ point $$(-3,-5)$$

$$x=−3$$

Exercise $$\PageIndex{79}$$

line $$x-2=0,$$ point $$(1,-2)$$

Exercise $$\PageIndex{80}$$

line $$x-6=0,$$ point $$(4,-3)$$

$$x=4$$

Exercise $$\PageIndex{81}$$

line $$y=5,$$ point $$(2,-2)$$

Exercise $$\PageIndex{82}$$

line $$y=1,$$ point $$(3,-4)$$

$$y=−4$$

Exercise $$\PageIndex{83}$$

line $$y+2=0,$$ point $$(3,-3)$$

Exercise $$\PageIndex{84}$$

line $$y+7=0,$$ point $$(1,-1)$$

$$y=−1$$

Find an Equation of a Line Perpendicular to a Given Line

In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

Exercise $$\PageIndex{85}$$

line $$y=-2 x+3,$$ point $$(2,2)$$

Exercise $$\PageIndex{86}$$

line $$y=-x+5,$$ point $$(3,3)$$

$$y=x$$

Exercise $$\PageIndex{87}$$

line $$y=\frac{3}{4} x-2,$$ point $$(-3,4)$$

Exercise $$\PageIndex{88}$$

line $$y=\frac{2}{3} x-4,$$ point $$(2,-4)$$

$$y=-\frac{3}{2} x-1$$

Exercise $$\PageIndex{89}$$

line $$2 x-3 y=8,$$ point $$(4,-1)$$

Exercise $$\PageIndex{90}$$

line $$4 x-3 y=5,$$ point $$(-3,2)$$

$$y=-\frac{3}{4} x-\frac{1}{4}$$

Exercise $$\PageIndex{91}$$

line $$2 x+5 y=6,$$ point $$(0,0)$$

Exercise $$\PageIndex{92}$$

line $$4 x+5 y=-3,$$ point $$(0,0)$$

$$y=\frac{5}{4} x$$

Exercise $$\PageIndex{93}$$

line $$y-3=0,$$ point $$(-2,-4)$$

Exercise $$\PageIndex{94}$$

line $$y-6=0,$$ point $$(-5,-3)$$

$$x=-5$$

Exercise $$\PageIndex{95}$$

line $$y$$-axis, point $$(3,4)$$

Exercise $$\PageIndex{96}$$

line $$y$$-axis, point $$(2,1)$$

$$y=1$$

Mixed Practice

In the following exercises, find the equation of each line. Write the equation in slope–intercept form.

Exercise $$\PageIndex{97}$$

Containing the points $$(4,3)$$ and $$(8,1)$$

Exercise $$\PageIndex{98}$$

Containing the points $$(2,7)$$ and $$(3,8)$$

$$y=x+5$$

Exercise $$\PageIndex{99}$$

$$m=\frac{1}{6},$$ containing point $$(6,1)$$

Exercise $$\PageIndex{100}$$

$$m=\frac{5}{6},$$ containing point $$(6,7)$$

$$y=\frac{5}{6} x+2$$

Exercise $$\PageIndex{101}$$

Parallel to the line $$4 x+3 y=6,$$ containing point $$(0,-3)$$

Exercise $$\PageIndex{102}$$

Parallel to the line $$2 x+3 y=6,$$ containing point $$(0,5)$$

$$y=-\frac{2}{3} x+5$$

Exercise $$\PageIndex{103}$$

$$m=-\frac{3}{4},$$ containing point $$(8,-5)$$

Exercise $$\PageIndex{104}$$

$$m=-\frac{3}{5},$$ containing point $$(10,-5)$$

$$y=-\frac{3}{5} x+1$$

Exercise $$\PageIndex{105}$$

Perpendicular to the line $$y-1=0,$$ point $$(-2,6)$$

Exercise $$\PageIndex{106}$$

Perpendicular to the line y-axis, point $$(-6,2)$$

$$y=2$$

Exercise $$\PageIndex{107}$$

Containing the points $$(4,3)$$ and $$(8,1)$$

Exercise $$\PageIndex{108}$$

Containing the points $$(-2,0)$$ and $$(-3,-2)$$

$$y=x+2$$

Exercise $$\PageIndex{109}$$

Parallel to the line $$x=-3,$$ containing point $$(-2,-1)$$

Exercise $$\PageIndex{110}$$

Parallel to the line $$x=-4,$$ containing point $$(-3,-5)$$

$$x=-3$$

Exercise $$\PageIndex{111}$$

Containing the points $$(-3,-4)$$ and $$(2,-5)$$

Exercise $$\PageIndex{112}$$

Containing the points $$(-5,-3)$$ and $$(4,-6)$$

$$y=-\frac{1}{3} x-\frac{14}{3}$$

Exercise $$\PageIndex{113}$$

Perpendicular to the line $$x-2 y=5,$$ containing point $$(-2,2)$$

Exercise $$\PageIndex{114}$$

Perpendicular to the line $$4 x+3 y=1,$$ containing point $$(0,0)$$

$$y=\frac{3}{4} x$$

Everyday Math

Exercise $$\PageIndex{115}$$

Cholesterol. The age, $$x,$$ and LDL cholesterol evel, $$y,$$ of two men are given by the points $$(18,68)$$ and $$(27,122) .$$ Find a linear equation that models the relationship between age and LDL cholesterol level.

Exercise $$\PageIndex{116}$$

Fuel consumption. The city mpg, $$x$$, and highway mpg, $$y,$$ of two cars are given by the points $$(29,40)$$ and $$(19,28) .$$ Find a
linear equation that models the relationship between city mpg and highway mp.

$$y=1.2 x+5.2$$

Writing Exercises

Exercise $$\PageIndex{117}$$

Why are all horizontal lines parallel?

Exercise $$\PageIndex{118}$$

Explain in your own words why the slopes of two perpendicular lines must have opposite signs.