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Mathematics LibreTexts

3.E: Review Exercises and Sample Exam

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    24126
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    Review Exercises

    Exercise \(\PageIndex{1}\) Rectangular Coordinate System

    Graph the given set of ordered pairs.

    1. \(\{(−3, 4), (−4, 0), (0, 3), (2, 4)\}\)
    2. \(\{(−5, 5), (−3, −1), (0, 0), (3, 2)\}\)
    3. Graph the points \((−3, 5), (−3, −3),\) and \((3, −3)\) on a rectangular coordinate plane. Connect the points and calculate the area of the shape.
    4. Graph the points \((−4, 1), (0, 1), (0, −2),\) and \((−4, −2)\) on a rectangular coordinate plane. Connect the points and calculate the area of the shape.
    5. Graph the points \((1, 0), (4, 0), (1, −5),\) and \((4, −5)\) on a rectangular coordinate plane. Connect the points and calculate the perimeter of the shape. 
    6. Graph the points \((−5, 2), (−5, −3), (1, 2),\) and \((1, −3)\) on a rectangular coordinate plane. Connect the points and calculate the perimeter of the shape.
    Answer

    1. 

    Screenshot (710).png

    Figure 3.E.1

    3. Area: \(24\) square units

    Screenshot (711).png

    Figure 3.E.2

    5. Perimeter: \(16\) units

    Screenshot (712).png

    Figure 3.E.3

    Exercise \(\PageIndex{2}\) Rectangular Coordinate System

    Calculate the distance between the given two points.

    1. \((−1, −2)\) and \((5, 6)\)
    2. \((2, −5)\) and \((−2, −2)\)
    3. \((−9, −3)\) and \((−8, 4)\)
    4. \((−1, 3)\) and \((1, −3)\)
    Answer

    1. \(10\) units

    3. \(5\sqrt{2}\) units

    Exercise \(\PageIndex{3}\) Rectangular Coordinate System

    Calculate the midpoint between the given points.

    1. \((−1, 3)\) and \((5, −7)\)
    2. \((6, −3)\) and \((−8, −11)\)
    3. \((7, −2)\) and \((−6, −1)\)
    4. \((−6, 0)\) and \((0, 0)\)
    5. Show algebraically that the points \((−1, −1), (1, −3),\) and \((2, 0)\) form an isosceles triangle.
    6. Show algebraically that the points \((2, −1), (6, 1),\) and \((5, 3)\) form a right triangle.
    Answer

    1. \((2,-2)\)

    3. \((\frac{1}{2},-\frac{3}{2})\)

    5. Answers may vary

    Exercise \(\PageIndex{4}\) Graph by Plotting Points

    Determine whether the given point is a solution.

    1. \(−5x+2y=7\); \((1, −1)\)
    2. \(6x−5y=4\); \((−1, −2)\)
    3. \(y=\frac{3}{4}x+1\); \((−\frac{2}{3}, \frac{1}{2})\)
    4. \(y=−\frac{3}{5}x−2\); \((10, −8)\)
    Answer

    1. No

    3. Yes

    Exercise \(\PageIndex{5}\) Graph by Plotting Points

    Find at least five ordered pair solutions and graph.

    1. \(y=−x+2\)
    2. \(y=2x−3\)
    3. \(y=\frac{1}{2}x−2\)
    4. \(y=−\frac{2}{3}x\)
    5. \(y=3\)
    6. \(x=−3\)
    7. \(x−5y=15\)
    8. \(2x−3y=12\)
    Answer

    1. 

    Screenshot (713).png

    Figure 3.E.4

    3.

    Screenshot (714).png

    Figure 3.E.5

    5.

    Screenshot (715).png

    Figure 3.E.6

    7.

    Screenshot (716).png

    Figure 3.E.7

    Exercise \(\PageIndex{6}\) Graph Using Intercepts

    Given the graph, find the \(x\)- and \(y\)- intercepts.

    1. 

    Screenshot (717).png

    Figure 3.E.8

    2.

    Screenshot (718).png

    Figure 3.E.9

    3.

    Screenshot (719).png

    Figure 3.E.10

    4.

    Screenshot (720).png

    Figure 3.E.11

    Answer

    1. \(y\)-intercept: \((0, −2)\); \(x\)-intercept: \((−4, 0)\)

    3. \(y\)-intercept: none; \(x\)-intercept: \((5, 0)\)

    Exercise \(\PageIndex{7}\) Graph Using Intercepts

    Find the intercepts and graph them.

    1. \(3x−4y=12\)
    2. \(2x−y=−4\)
    3. \(\frac{1}{2}x−\frac{1}{3}y=1\)
    4. \(−\frac{1}{2}x+\frac{2}{3}y=2\)
    5. \(y=−\frac{5}{3}x+5\)
    6. \(y=−3x+4\)
    Answer

    1. 

    Screenshot (721).png

    Figure 3.E.12

    3.

    Screenshot (722).png

    Figure 3.E.13

    5.

    Screenshot (723).png

    Figure 3.E.14

    Exercise \(\PageIndex{8}\) Graph Using the \(y\)-Intercept and Slope

    Given the graph, determine the slope and \(y\)-intercept.

    1.

    Screenshot (724).png

    Figure 3.E.15

    2.

    Screenshot (725).png

    Figure 3.E.16

    Answer

    1. \(y\)-intercept: \((0, 1)\); slope: \(−2\)

    Exercise \(\PageIndex{9}\) Graph Using the \(y\)-Intercept and Slope

    Determine the slope, given two points.

    1. \((−3, 8)\) and \((5, −6)\)
    2. \((0, −5)\) and \((−6, 3)\)
    3. \((\frac{1}{2}, −\frac{2}{3})\) and \((\frac{1}{4}, −\frac{1}{3})\)
    4. \((5, −\frac{3}{4})\) and \((2, −\frac{3}{4})\)
    Answer

    1. \(-\frac{7}{4}\)

    3. \(-\frac{4}{3}\)

    Exercise \(\PageIndex{10}\) Graph Using the \(y\)-Intercept and Slope

    Express in slope-intercept form and identify the slope and \(y\)-intercept.

    1. \(12x−4y=8\)
    2. \(3x−6y=24\)
    3. \(−\frac{1}{3}x+\frac{3}{4}y=1\)
    4. \(−5x+3y=0\)
    Answer

    1. \(y=3x−2\); slope: \(3\); \(y\)-intercept \((0, −2)\)

    3. \(y=\frac{4}{9}x+\frac{4}{3}\); slope: \(\frac{4}{9}\); \(y\)-intercept \((0, \frac{4}{3})\)

    Exercise \(\PageIndex{11}\) Graph Using the \(y\)-Intercept and Slope

    1. \(y=−x+3\)
    2. \(y=4x−1\)
    3. \(y=−2x\)
    4. \(y=−\frac{5}{2}x+3\)
    5. \(2x−3y=9\)
    6. \(2x+\frac{3}{2}y=3\)
    7. \(y=0\)
    8. \(x−4y=0\)
    Answer

    1. 

    Screenshot (726).png

    Figure 3.E.17

    3.

    Screenshot (727).png

    Figure 3.E.18

    5.

    Screenshot (728).png

    Figure 3.E.19

    7.

    Screenshot (729).png

    Figure 3.E.20

    Exercise \(\PageIndex{12}\) Finding Linear Equations

    Given the graph, determine the equation of the line.

    1.

    Screenshot (730).png

    Figure 3.E.21

    2.

    Screenshot (731).png

    Figure 3.E.22

    3.

    Screenshot (732).png

    Figure 3.E.23

    4.

    Screenshot (733).png

    Figure 3.E.24

    Answer

    1. \(y=−2x+1\)

    3. \(y=−5\)

    Exercise \(\PageIndex{13}\) Finding Linear Equations

    Find the equation of a line, given the slope and a point on the line.

    1. \(m = \frac{1}{2}\); \((−4, 8)\)
    2. \(m = −\frac{1}{5}\); \((−5, −9)\)
    3. \(m = \frac{2}{3}\); \((1, −2)\)
    4. \(m = −\frac{3}{4}\); \((2, −3)\)
    Answer

    1. \(y=\frac{1}{2}x+10\)

    3. \(y=\frac{2}{3}x−\frac{8}{3}\)

    Exercise \(\PageIndex{14}\) Finding Linear Equations

    Find the equation of the line given two points on the line.

    1. \((−5, −5)\) and \((10, 7)\)
    2. \((−6, 12)\) and \((3, −3)\)
    3. \((2, −1)\) and \((−2, 2)\)
    4. \((\frac{5}{2}, −2)\) and \((−5, \frac{5}{2})\)
    5. \((7, −6)\) and \((3, −6)\)
    6. \((10, 1)\) and \((10, −3)\)
    Answer

    1. \(y=\frac{4}{5}x−1\)

    3. \(y=−\frac{3}{4}x+\frac{1}{2}\)

    5. \(y=−6\)

    Exercise \(\PageIndex{15}\) Parallel and Perpendicular Lines

    Determine if the lines are parallel, perpendicular, or neither.

    1. \(\left\{\begin{aligned}−3x+7y&=14\\6x−14y&=42\end{aligned}\right.\)
    2. \(\left\{\begin{aligned}2x+3y&=18\\2x−3y&=36\end{aligned}\right.\)
    3. \(\left\{\begin{aligned}x+4y&=2\\8x−2y=&−1\end{aligned}\right.\)
    4. \(\left\{\begin{aligned}y&=2\\x&=2\end{aligned}\right.\)
    Answer

    1. Parallel

    3. Perpendicular

    Exercise \(\PageIndex{16}\) Parallel and Perpendicular Lines

    Find the equation of the line in slope-intercept form.

    1. Parallel to \(5x−y=15\) and passing through \((−10, −1)\).
    2. Parallel to \(x−3y=1\) and passing through \((2, −2)\).
    3. Perpendicular to \(8x−6y=4\) and passing through \((8, −1)\).
    4. Perpendicular to \(7x+y=14\) and passing through \((5, 1)\).
    5. Parallel to \(y=1\) and passing through \((4, −1)\).
    6. Perpendicular to \(y=1\) and passing through \((4, −1)\).
    Answer

    1. \(y=5x+49\)

    3. \(y=−\frac{3}{4}x+5\)

    5. \(y=−1\)

    Exercise \(\PageIndex{17}\) Introduction to Functions

    Determine the domain and range and state whether it is a function or not.

    1. \(\{(−10, −1), (−5, 2), (5, 2)\}\)

    2. \(\{(−12, 4), (−1, −3), (−1, −2)\}\)

    3. 

    Screenshot (734).png

    Figure 3.E.25

    4.

    Screenshot (735).png

    Figure 3.E.26

    5.

    Screenshot (736).png

    Figure 3.E.27

    6.

    Screenshot (737).png

    Figure 3.E.28

    Answer

    1. Domain: \(\{−10, −5, 5\}\); range: \(\{−1, 2\}\); function: yes

    3. Domain: \(R\); range: \(R\); function: yes

    5. Domain: \([−3,∞)\); range: \(R\); function: no

    Exercise \(\PageIndex{18}\) Introduction to Functions

    Given the following,

    1. \(f(x)=9x−4\), find \(f(−1)\).
    2. \(f(x)=−5x+1\), find \(f(−3)\).
    3. \(g(x)=\frac{1}{2}x−\frac{1}{3}\), find \(g(−\frac{1}{3})\).
    4. \(g(x)=−\frac{3}{4}x+\frac{1}{3}\), find \(g(\frac{2}{3})\).
    5. \(f(x)=9x−4\), find \(x\) when \(f(x)=0\).
    6. \(f(x)=−5x+1\), find \(x\) when \(f(x)=2\).
    7. \(g(x)=\frac{1}{2}x−\frac{1}{3}\), find \(x\) when \(g(x)=1\).
    8. \(g(x)=−\frac{3}{4}x+\frac{1}{3}\), find \(x\) when \(g(x)=−1\).
    Answer

    1. \(f(−1)=−13\)

    3. \(g(−\frac{1}{3})=−\frac{1}{2}\)

    5. \(x=\frac{4}{9}\)

    7. \(x=\frac{8}{3}\)

    Exercise \(\PageIndex{19}\) Introduction to Functions

    Given the graph of a function \(f(x)\), determine the following.

    Screenshot (738).png

    Figure 3.E.29

    1. \(f(3)\)
    2. \(x\) when \(f(x)=4\)
    Answer

    1. \(f(3)=−2\)

    Exercise \(\PageIndex{20}\) Linear Inequalities (Two Variables)

    Is the ordered pair a solution to the given inequality?

    1. \(6x−2y≤1\); \((−3, −7)\)
    2. \(−3x+y>2\); \((0, 2)\)
    3. \(6x−10y<-1\); \((5,-3)\)
    4. \(x-\frac{1}{3}y>0\); \((1, 4)\)
    5. \(y>0\); \((−3, −1)\)
    6. \(x≤−5\); \((−6, 4)\) 
    Answer

    1. Yes

    3. No

    5. Yes

    Exercise \(\PageIndex{21}\) Linear Inequalities (Two Variables)

    Graph the solution set.

    1. \(y≥−2x+1\)
    2. \(y<3x−4\)
    3. \(−x+y≤3\)
    4. \(\frac{5}{2}x+\frac{1}{2}y≤2\)
    5. \(3x−5y>0\)
    6. \(y>0\)
    Answer

    1. 

    Screenshot (739).png

    Figure 3.E.30

    3.

    Screenshot (740).png

    Figure 3.E.31

    5.

    Screenshot (741).png

    Figure 3.E.32

    Sample Exam

    Exercise \(\PageIndex{22}\)

    1. Graph the points \((−4, −2), (−4, 1),\) and \((0, −2)\) on a rectangular coordinate plane. Connect the points and calculate the area of the shape.
    2. Is \((−2, 4)\) a solution to \(3x−4y=−10\)? Justify your answer.
    Answer

    1. Area: \(6\) square units

    Screenshot (742).png

    Figure 3.E.33

    Exercise \(\PageIndex{23}\)

    Given the set of \(x\)-values \(\{−2, −1, 0, 1, 2\}\), find the corresponding \(y\)-values and graph the following.

    1. \(y=x−1\)
    2. \(y=−x+1\)
    3. On the same set of axes, graph \(y=4\) and \(x=−3\). Give the point where they intersect.
    Answer

    1. 

    Screenshot (743).png

    Figure 3.E.34

    3. Intersection: \((-3,4)\)

    Screenshot (744).png

    Figure 3.E.35

    Exercise \(\PageIndex{24}\)

    Find the \(x\)- and \(y\)-intercepts and use those points to graph the following.

    1. \(2x−y=8\)
    2. \(12x+5y=15\)
    3. Calculate the slope of the line passing through \((−4, −5)\) and \((−3, 1)\).
    Answer

    2. 

    Screenshot (745).png

    Figure 3.E.36

    Exercise \(\PageIndex{25}\)

    Determine the slope and \(y\)-intercept. Use them to graph the following.

    1. \(y=−\frac{3}{2}x+6\)
    2. \(5x−2y=6\)
    3. Given \(m=−3\), determine \(m_{⊥}\).
    4. Are the given lines parallel, perpendicular, or neither? \(\left\{\begin{aligned} -2x+3y&=-12\\4x-6y&=30 \end{aligned}\right.\)
    5. Determine the slope of the given lines.
      1. \(y=−2\)
      2. \(x=\frac{1}{3}\)
      3. Are these lines parallel, perpendicular, or neither?
    6. Determine the equation of the line with slope \(m=−\frac{3}{4}\) passing through \((8, 1)\).
    7. Find the equation to the line passing through \((−2, 3)\) and \((4, 1)\).
    8. Find the equation of the line parallel to \(5x−y=6\) passing through \((−1, −2)\).
    9. Find the equation of the line perpendicular to \(−x+2y=4\) passing through \((\frac{1}{2}, 5)\).
    Answer

    1. Slope: \(−\frac{3}{2}\); \(y\)-intercept: \((0, 6)\)

    Screenshot (746).png

    Figure 3.E.37

    3. \(m_{⊥}=\frac{1}{3}\)

    5. a. \(0\); b. Undefined; c. Perpendicular

    6. \(y=−\frac{3}{4}x+7\)

    8. \(y=5x+3\)

    Exercise \(\PageIndex{26}\)

    Given a linear function \(f(x)=−\frac{4}{5}x+2\), determine the following.

    1. \(f(10)\)
    2. \(x\) when \(f(x)=0\)
    3. Graph the solution set: \(3x−4y>4\).
    4. Graph the solution set: \(y−2x≥0\).
    5. A rental car company charges $\(32.00\) plus $\(0.52\) per mile driven. Write an equation that gives the cost of renting the car in terms of the number of miles driven. Use the formula to determine the cost of renting the car and driving it \(46\) miles.
    6. A car was purchased new for $\(12,000\) and was sold 5 years later for $\(7,000\). Write a linear equation that gives the value of the car in terms of its age in years.
    7. The area of a rectangle is \(72\) square meters. If the width measures \(4\) meters, then determine the length of the rectangle.
    Answer

    1. \(f(10)=−6\)

    3. 

    Screenshot (747).png

    Figure 3.E.38

    5. cost\(=0.52x+32\); $\(55.92\)

    7. \(18\) meters