7.10: Exercise Supplement

Last updated
Save as PDF

Page ID: 60041

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

Exercise Supplement

Graphing Linear Equations and Inequalities in One Variable

For the following problems, graph the equations and inequalities.

Exercise $\PageIndex{1}$

$6x−18=6$

$A horizontal line with arrows on both ends labeled as x.$

Answer

$x=4$

$A number line with arrows on each end, labeled from negative two to four in increments of one. There is a closed circle at four.$

Exercise $\PageIndex{2}$

$4x−3=−7$

$A horizontal line with arrows on both ends labeled as x.$

Exercise $\PageIndex{3}$

$5x−1=2$

$A horizontal line with arrows on both ends labeled as x.$

Answer

$x = \dfrac{3}{5}$

$A number line with arrows on each end, labeled from negative one to ttwo in increments of one. There is a closed circle at three over five.$

Exercise $\PageIndex{4}$

$10x−16<4$

$A horizontal line with arrows on both ends labeled as x.$

Exercise $\PageIndex{5}$

$−2y+1≤5$

$A horizontal line with arrows on both ends labeled as y.$

Answer: $y≥−2$

Exercise $\PageIndex{6}$

$\dfrac{-7a}{12} \ge 2$

$A horizontal line with arrows on both ends labeled as a.$

Exercise $\PageIndex{7}$

$3x+4≤12$

$A horizontal line with arrows on both ends labeled as x.$

Answer

$x \le \dfrac{8}{3}$

$A number line with arrows on each end, labeled from negative two to three, in increments of one. There is a closed circle at a point between two and three. A dark line is orginating from this circle and heading towards the left of it.$

Exercise $\PageIndex{8}$

$−16≤5x−1≤−11$

$A horizontal line with arrows on both ends labeled as x.$

Exercise $\PageIndex{9}$

$0<−3y+9≤9$

$A horizontal line with arrows on both ends labeled as y.$

Answer

$0≤y<3$

$A number line with arrows on each end, labeled from negative one to four, in increments of one. There is a closed circle at zero and an open circle at three. These circles are connected by a a black line.$

Exercise $\PageIndex{10}$

$\dfrac{-5c}{2} + 1 = 7$

$A horizontal line with arrows on both ends labeled as c.$

Plotting Points in the Plane

Exercise $\PageIndex{11}$

Draw a coordinate system and plot the following ordered pairs.

$(3, 1), (4, -2), (-1, -3), (0, 3), (3, 0), (5, -\dfrac{2}{3})$

Answer: $Total six points plotted in an xy-coordinate plane. The coordinates of these points are negative one, negative three; zero, three; three, one; three, zero; four, negative two; and five, negative two over three.$

Exercise $\PageIndex{12}$

As accurately as possible, state the coordinates of the points that have been plotted on the graph.

$Total seven points plotted on an xy-plane. The coordinates of these points are one, three; two, one; three,zero; three, negative two; negative one, negative three; negative three, three.$

Graphing Linear Equations in Two Variables

Exercise $\PageIndex{13}$

What is the geometric structure of the graph of all the solutions to the linear equation $y=4x−9$?

Answer: a straight line

Graphing Linear Equations in Two Variables - Graphing Equations in Slope-Intercept Form

For the following problems, graph the equations.

Exercise $\PageIndex{14}$

$y−x=2$

Exercise $\PageIndex{15}$

$y+x−3=0$

Answer: $A graph of a line passing through two points with coordinates zero, three and five, zero.$

Exercise $\PageIndex{16}$

$−2x+3y=−6$

Exercise $\PageIndex{17}$

$2y+x−8=0$

Answer: $A graph of a line passing through two points with coordinates zero, four and eight, zero.$

Exercise $\PageIndex{18}$

$4(x−y)=12$

Exercise $\PageIndex{19}$

$3y−4x+12=0$

Answer: $A graph of a line passing through two points with coordinates zero, three and negative four, zero.$

Exercise $\PageIndex{20}$

$y=−3$

Exercise $\PageIndex{21}$

$y−2=0$

Answer: $A graph of a line parallel to x-axis in an xy plane. The line is labeled as ' y equals two'. The line crosses the y-axis at y equals two.$

Exercise $\PageIndex{22}$

$x=4$

Exercise $\PageIndex{23}$

$x+1=0$

Answer: $A graph of a line parallel to y-axis in an xy plane. The line is labeled as 'x equals negative one'. The line crosses the x-axis at x equals negative one.$

Exercise $\PageIndex{24}$

$x=0$

Exercise $\PageIndex{25}$

$y=0$

Answer: $A graph of a line in an xy plane coincident to x-axis labeled as 'y equals zero'.$

The Slope-Intercept Form of a Line

Exercise $\PageIndex{26}$

Write the slope-intercept form of a straight line.

Exercise $\PageIndex{27}$

The slope of a straight line is a ____ of the steepness of the line.

Answer: measure

Exercise $\PageIndex{28}$

Write the formula for the slope of a line that passes through the points $(x_1,y_1)$ and $(x_2,y_2)$.

For the following problems, determine the slope and y-intercept of the lines.

Exercise $\PageIndex{29}$

$y=4x+10$

Answer

slope: $4$

$y$-intercept: $(0,10)$

Exercise $\PageIndex{30}$

$y=3x−11$

Exercise $\PageIndex{31}$

$y=9x−1$

Answer

slope: $9$

$y$-intercept: $(0,-1)$

Exercise $\PageIndex{32}$

$y=−x+2$

Exercise $\PageIndex{33}$

$y=−5x−4$

Answer

slope: $-5$

$y$-intercept: $(0,-4)$

Exercise $\PageIndex{34}$

$y=x$

Exercise $\PageIndex{35}$

$y=−6x$

Answer

slope: $-6$

$y$-intercept: $(0,0)$

Exercise $\PageIndex{36}$

$3y=4x+9$

Exercise $\PageIndex{37}$

$4y=5x+1$

Answer

slope: $\dfrac{5}{4}$

$y$-intercept: $(0,\dfrac{1}{4})$

Exercise $\PageIndex{38}$

$2y=9x$

Exercise $\PageIndex{39}$

$5y+4x=6$

Answer

slope: $-\dfrac{4}{5}$

$y$-intercept: $(0,\dfrac{6}{5})$

Exercise $\PageIndex{40}$

$7y+3x=10$

Exercise $\PageIndex{41}$

$6y−12x=24$

Answer

slope: $2$

$y$-intercept: $(0,4)$

Exercise $\PageIndex{42}$

$5y−10x−15=0$

Exercise $\PageIndex{43}$

$3y+3x=1$

Answer

slope: $-1$

$y$-intercept: $(0,\dfrac{1}{3})$

Exercise $\PageIndex{44}$

$7y+2x=0$

Exercise $\PageIndex{45}$

$y=4$

Answer

slope: $0$

$y$-intercept: $(0,4)$

For the following problems, find the slope, if it exists, of the line through the given pairs of points.

Exercise $\PageIndex{46}$

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

Exercise $\PageIndex{47}$

$(8,−2),(10,−6)$

Answer: slope: $−2$

Exercise $\PageIndex{48}$

$(0,5),(3,4)$

Exercise $\PageIndex{49}$

$(1,−4),(3,3)$

Answer: slope: $\dfrac{7}{2}$

Exercise $\PageIndex{50}$

$(0,0),(−8,−5)$

Exercise $\PageIndex{51}$

$(−6,1),(−2,7)$

Answer: slope: $\dfrac{3}{2}$

Exercise $\PageIndex{52}$

$(−3,−2),(−4,−5)$

Exercise $\PageIndex{53}$

$(4,7),(4,−2)$

Answer: No Slope

Exercise $\PageIndex{54}$

$(−3,1),(4,1)$

Exercise $\PageIndex{55}$

$(\dfrac{1}{3}, \dfrac{3}{4}), (\dfrac{2}{9}, -\dfrac{5}{6})$

Answer: slope: $\dfrac{57}{4}$

Exercise $\PageIndex{56}$

Moving left to right, lines with slope rise while lines with slope decline.

Exercise $\PageIndex{57}$

Compare the slopes of parallel lines.

Answer: The slopes of parallel lines are equal.

Finding the Equation of a Line

For the following problems, write the equation of the line using the given information. Write the equation in slope-intercept form.

Exercise $\PageIndex{58}$

Slope=$4$, $y$-intercept=$5$

Exercise $\PageIndex{59}$

Slope=$3$, $y$-intercept=$-6$

Answer: $y=3x−6$

Exercise $\PageIndex{60}$

Slope=$1$, $y$-intercept=$8$

Exercise $\PageIndex{61}$

Slope=$1$, $y$-intercept=$-2$

Answer: $y=x−2$

Exercise $\PageIndex{62}$

Slope=$-5$, $y$-intercept=$1$

Exercise $\PageIndex{63}$

Slope=$-11$, $y$-intercept=$-4$

Answer: $y=−11x−4$

Exercise $\PageIndex{64}$

Slope=$2$, $y$-intercept=$0$

Exercise $\PageIndex{65}$

Slope=$-1$, $y$-intercept=$0$

Answer: $y=−x$

Exercise $\PageIndex{66}$

$m=3,(4,1)$

Exercise $\PageIndex{67}$

$m=2,(1,5)$

Answer: $y=2x+3$

Exercise $\PageIndex{68}$

$m=6,(5,−2)$

Exercise $\PageIndex{69}$

$m=−5,(2,−3)$

Answer: $y=−5x+7$

Exercise $\PageIndex{70}$

$m=−9,(−4,−7)$

Exercise $\PageIndex{71}$

$m=−2,(0,2)$

Answer: $y=−2x+2$

Exercise $\PageIndex{72}$

$m=−1,(2,0)$

Exercise $\PageIndex{73}$

$(2,3),(3,5)$

Answer: $y=2x−1$

Exercise $\PageIndex{74}$

$(4,4),(5,1)$

Exercise $\PageIndex{75}$

$(6,1),(5,3)$

Answer: $y=−2x+13$

Exercise $\PageIndex{76}$

$(8,6),(7,2)$

Exercise $\PageIndex{77}$

$(−3,1),(2,3)$

Answer: $y = \dfrac{2}{5}x + \dfrac{11}{5}$

Exercise $\PageIndex{78}$

$(−1,4),(−2,−4)$

Exercise $\PageIndex{79}$

$(0,−5),(6,−1)$

Answer: $y = \dfrac{2}{3}x - 5$

Exercise $\PageIndex{80}$

$(2,1),(6,1)$

Exercise $\PageIndex{81}$

$(−5,7),(−2,7)$

Answer: $y=7$ (zero slope)

Exercise $\PageIndex{82}$

$(4,1),(4,3)$

Exercise $\PageIndex{83}$

$(−1,−1),(−1,5)$

Answer: $x=−1$ (no slope)

Exercise $\PageIndex{84}$

$(0,4),(0,−3)$

Exercise $\PageIndex{85}$

$(0,2),(1,0)$

Answer: $y=−2x+2$

For the following problems, reading only from the graph, determine the equation of the line.

Exercise $\PageIndex{86}$

$A graph of a line sloped up and to the right. The line crosses the y-axis at y equals one, and crosses the x-axis at x equals negative two.$

Exercise $\PageIndex{87}$

$A graph of a line sloped up and to the right. The line crosses the x-axis at x equals three, and crosses the y-axis at y equals negative two.$

Answer: $y = \dfrac{2}{3}x - 2$

Exercise $\PageIndex{88}$

$A graph of a line sloped down and to the right. The line crosses the y-axis at y equals one, and crosses the x-axis at x equals four.$

Exercise $\PageIndex{89}$

$A graph of a parallel to x-axis. The line crosses the y-axis at y equals negative two.$

Answer: $y=−2$

Exercise $\PageIndex{90}$

$A graph of a parallel to y-axis. The line crosses the x-axis at x equals three.$

Exercise $\PageIndex{91}$

$A graph of a parallel to x-axis. The line crosses the y-axis at y equals one.$

Answer: $y=1$