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Mathematics LibreTexts

4.6E: Exercises

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    30401
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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 \((0,1)\)

    Answer

    \(y=4x+1\)

    Exercise \(\PageIndex{3}\)

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

    Exercise \(\PageIndex{4}\)

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

    Answer

    \(y=8x−6\)

    Exercise \(\PageIndex{5}\)

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

    Exercise \(\PageIndex{6}\)

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

    Answer

    \(y=−x+7\)

    Exercise \(\PageIndex{7}\)

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

    Exercise \(\PageIndex{8}\)

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

    Answer

    \(y=−3x−1\)

    Exercise \(\PageIndex{9}\)

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

    Exercise \(\PageIndex{10}\)

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

    Answer

    \(y=\frac{1}{5} x-5\)

    Exercise \(\PageIndex{11}\)

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

    Exercise \(\PageIndex{12}\)

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

    Answer

    \(y=-\frac{2}{3} x-3\)

    Exercise \(\PageIndex{13}\)

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

    Exercise \(\PageIndex{14}\)

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

    Answer

    \(y=2\)

    Exercise \(\PageIndex{15}\)

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

    Exercise \(\PageIndex{16}\)

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

    Answer

    \(y=−4x\)

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

    Exercise \(\PageIndex{17}\)

    The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (1, negative 2) is plotted. A line intercepts the y-axis at (0, negative 5), passes through the point (1, negative 2), and intercepts the x-axis at (5 thirds, 0).

    Exercise \(\PageIndex{18}\)

    The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (2, 0) is plotted. A line intercepts the y-axis at (0, 4) and intercepts the x-axis at (2, 0).

    Answer

    \(y=−2x+4\)

    Exercise \(\PageIndex{19}\)

    The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (6, 0) is plotted. A line intercepts the y-axis at (0, negative 3) and intercepts the x-axis at (6, 0).

    Exercise \(\PageIndex{20}\)

    The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (4, 5) is plotted. A line intercepts the x-axis at (negative 8 thirds, 0), intercepts the y-axis at (0, 2), and passes through the point (4, 5).

    Answer

    \(y=\frac{3}{4} x+2\)

    Exercise \(\PageIndex{21}\)

    The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (3, negative 1) is plotted. A line intercepts the y-axis at (0, 2), intercepts the x-axis at (9 fourths, 0), and passes through the point (3, negative 1).

    Exercise \(\PageIndex{22}\)

    The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (2, negative 4) is plotted. A line intercepts the x-axis at (negative 2 thirds, 0), intercepts the y-axis at (0, negative 1), and passes through the point (2, negative 4).

    Answer

    \(y=-\frac{3}{2} x-1\)

    Exercise \(\PageIndex{23}\)

    The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (2, negative 2) is plotted. A line running parallel to the x-axis intercepts the y-axis at (0, negative 2) and passes through the point (2, negative 2).

    Exercise \(\PageIndex{24}\)

    The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (negative 3, 6) is plotted. A line running parallel to the x-axis passes through (negative 3, 6) and intercepts the y-axis at (0, 6).

    Answer

    \(y=6\)

    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 \((8,2)\)

    Answer

    \(y=\frac{3}{8} x-1\)

    Exercise \(\PageIndex{27}\)

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

    Exercise \(\PageIndex{28}\)

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

    Answer

    \(y=\frac{5}{6} x+2\)

    Exercise \(\PageIndex{29}\)

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

    Exercise \(\PageIndex{30}\)

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

    Answer

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

    Exercise \(\PageIndex{31}\)

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

    Exercise \(\PageIndex{32}\)

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

    Answer

    \(y=-\frac{1}{3} x-11\)

    Exercise \(\PageIndex{33}\)

    Horizontal line containing \((−2,5)\)

    Exercise \(\PageIndex{34}\)

    Horizontal line containing \((−1,4)\)

    Answer

    \(y=4\)

    Exercise \(\PageIndex{35}\)

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

    Exercise \(\PageIndex{36}\)

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

    Answer

    \(y=−7\)

    Exercise \(\PageIndex{37}\)

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

    Exercise \(\PageIndex{38}\)

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

    Answer

    \(y=-\frac{5}{2} x-22\)

    Exercise \(\PageIndex{39}\)

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

    Exercise \(\PageIndex{40}\)

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

    Answer

    \(y=-4 x-11\)

    Exercise \(\PageIndex{41}\)

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

    Exercise \(\PageIndex{42}\)

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

    Answer

    \(y=−8\)

    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 \((2,5)\)

    Answer

    \(y=−4x+13\)

    Exercise \(\PageIndex{45}\)

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

    Exercise \(\PageIndex{46}\)

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

    Answer

    \(y=x+5\)

    Exercise \(\PageIndex{47}\)

    \((−3,−4)\) and \((5−2)\)

    Exercise \(\PageIndex{48}\)

    \((−5,−3)\) and \((4,−6)\)

    Answer

    \(y=-\frac{1}{3} x-\frac{14}{3}\)

    Exercise \(\PageIndex{49}\)

    \((−1,3)\) and \((−6,−7)\)

    Exercise \(\PageIndex{50}\)

    \((−2,8)\) and \((−4,−6)\)

    Answer

    \(y=7x+22\)

    Exercise \(\PageIndex{51}\)

    \((6,−4)\) and \((−2,5)\)

    Exercise \(\PageIndex{52}\)

    \((3,−2)\) and \((−4,4)\)

    Answer

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

    Exercise \(\PageIndex{53}\)

    \((0,4)\) and \((2,−3)\)

    Exercise \(\PageIndex{54}\)

    \((0,−2)\) and \((−5,−3)\)

    Answer

    \(y=\frac{1}{5} x-2\)

    Exercise \(\PageIndex{55}\)

    \((7,2)\) and \((7,−2)\)

    Exercise \(\PageIndex{56}\)

    \((4,2)\) and \((4,−3)\)

    Answer

    \(x=4\)

    Exercise \(\PageIndex{57}\)

    \((−7,−1)\) and \((−7,−4)\)

    Exercise \(\PageIndex{58}\)

    \((−2,1)\) and \((−2,−4)\)

    Answer

    \(x=−2\)

    Exercise \(\PageIndex{59}\)

    \((6,1)\) and \((0,1)\)

    Exercise \(\PageIndex{60}\)

    \((6,2)\) and \((−3,2)\)

    Answer

    \(y=2\)

    Exercise \(\PageIndex{61}\)

    \((3,−4)\) and \((5,−4)\)

    Exercise \(\PageIndex{62}\)

    \((−6,−3)\) and \((−1,−3)\)

    Answer

    \(y=−3\)

    Exercise \(\PageIndex{63}\)

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

    Exercise \(\PageIndex{64}\)

    \((0,0)\) and \((1,4)\)

    Answer

    \(y=4x\)

    Exercise \(\PageIndex{65}\)

    \((−2,−3)\) and \((−5,−6)\)

    Exercise \(\PageIndex{66}\)

    \((−3,0)\) and \((−7,−2)\)

    Answer

    \(y=\frac{1}{2} x+\frac{3}{2}\)

    Exercise \(\PageIndex{67}\)

    \((8,−1)\) and \((8,−5)\)

    Exercise \(\PageIndex{68}\)

    \((3,5)\) and \((−7,5)\)

    Answer

    \(y=5\)

    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 \((2,5)\)

    Answer

    \(y=3 x-1\)

    Exercise \(\PageIndex{71}\)

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

    Exercise \(\PageIndex{72}\)

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

    Answer

    \(y=−3x+3\)

    Exercise \(\PageIndex{73}\)

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

    Exercise \(\PageIndex{74}\)

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

    Answer

    \(y=2x−6\)

    Exercise \(\PageIndex{75}\)

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

    Exercise \(\PageIndex{76}\)

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

    Answer

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

    Exercise \(\PageIndex{77}\)

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

    Exercise \(\PageIndex{78}\)

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

    Answer

    \(x=−3\)

    Exercise \(\PageIndex{79}\)

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

    Exercise \(\PageIndex{80}\)

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

    Answer

    \(x=4\)

    Exercise \(\PageIndex{81}\)

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

    Exercise \(\PageIndex{82}\)

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

    Answer

    \(y=−4\)

    Exercise \(\PageIndex{83}\)

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

    Exercise \(\PageIndex{84}\)

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

    Answer

    \(y=−1\)

    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 \((3,3)\)

    Answer

    \(y=x\)

    Exercise \(\PageIndex{87}\)

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

    Exercise \(\PageIndex{88}\)

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

    Answer

    \(y=-\frac{3}{2} x-1\)

    Exercise \(\PageIndex{89}\)

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

    Exercise \(\PageIndex{90}\)

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

    Answer

    \(y=-\frac{3}{4} x-\frac{1}{4}\)

    Exercise \(\PageIndex{91}\)

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

    Exercise \(\PageIndex{92}\)

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

    Answer

    \(y=\frac{5}{4} x\)

    Exercise \(\PageIndex{93}\)

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

    Exercise \(\PageIndex{94}\)

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

    Answer

    \(x=-5\)

    Exercise \(\PageIndex{95}\)

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

    Exercise \(\PageIndex{96}\)

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

    Answer

    \(y=1\)

    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 \((3,8)\)

    Answer

    \(y=x+5\)

    Exercise \(\PageIndex{99}\)

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

    Exercise \(\PageIndex{100}\)

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

    Answer

    \(y=\frac{5}{6} x+2\)

    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 \((0,5)\)

    Answer

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

    Exercise \(\PageIndex{103}\)

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

    Exercise \(\PageIndex{104}\)

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

    Answer

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

    Exercise \(\PageIndex{105}\)

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

    Exercise \(\PageIndex{106}\)

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

    Answer

    \(y=2\)

    Exercise \(\PageIndex{107}\)

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

    Exercise \(\PageIndex{108}\)

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

    Answer

    \(y=x+2\)

    Exercise \(\PageIndex{109}\)

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

    Exercise \(\PageIndex{110}\)

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

    Answer

    \(x=-3\)

    Exercise \(\PageIndex{111}\)

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

    Exercise \(\PageIndex{112}\)

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

    Answer

    \(y=-\frac{1}{3} x-\frac{14}{3}\)

    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 \((0,0)\)

    Answer

    \(y=\frac{3}{4} x\)

    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.

    Answer

    \(y=1.2 x+5.2\)

    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.

    Answer

    Answers will vary.

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “find the equation of the line given the slope and y-intercept,”, “find an equation of the line given the slope and a point,” “find an equation of the line given two points,” “find an equation of a line parallel to a given line,” and “find an equation of a line perpendicular to a given line.” The rest of the cells are blank.

    b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?