Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

4.E: Review Exercises and Sample Exam

  • Page ID
    18352
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Review Exercises

    Exercise \(\PageIndex{1}\) Solving Linear Systems by Graphing

    Determine whether the given ordered pair is a solution to the given system.

    1. \((1,-3)\); \(\left\{\begin{aligned} 5x−y&=8\\−3x+y&=−6 \end{aligned}\right.\)
    2. \((-3,-4)\); \(\left\{\begin{aligned} 4x−12y&=−10\\6x−5y&=−2 \end{aligned}\right.\)
    3. \((-1,\frac{1}{5})\); \(\left\{\begin{aligned} \frac{3}{5}x-\frac{1}{3}y&=-\frac{2}{3}\\-\frac{1}{5}x-\frac{1}{2}y&=\frac{1}{10} \end{aligned}\right.\)
    4. \((\frac{1}{2},-1)\); \(\left\{\begin{aligned} x+\frac{3}{4}y&=-\frac{1}{4}\\ \frac{2}{3}x-y&=\frac{4}{3} \end{aligned}\right.\)
    Answer

    1. Yes

    3. Yes

    Exercise \(\PageIndex{2}\) Solving Linear Systems by Graphing

    Given the graph, determine the simultaneous solution.

    1.

    Screenshot (448).png
    Figure 4.E.1

    2.

    Screenshot (449).png
    Figure 4.E.2

    3.

    Screenshot (450).png
    Figure 4.E.3

    4.

    Screenshot (451).png
    Figure 4.E.4
    Answer

    1. \((−3, 1)\)

    3. \(Ø\)

    Exercise \(\PageIndex{3}\) Solving Linear Systems by Graphing

    Solve by graphing.

    1. \(\left\{\begin{aligned} y&=12x-3\\y&=-\frac{3}{4}x+2 \end{aligned}\right.\)
    2. \(\left\{\begin{aligned} y&=5\\y&=-\frac{4}{5}x+1 \end{aligned}\right.\)
    3. \(\left\{\begin{aligned} x-2y&=0\\2x-3y&=3 \end{aligned}\right.\)
    4. \(\left\{\begin{aligned} 5x-y&=-11\\-4x+2y&=16 \end{aligned}\right.\)
    5. \(\left\{\begin{aligned} \frac{5}{2}x+2y&=6\\5x+4y&=12 \end{aligned}\right.\)
    6. \(\left\{\begin{aligned} 6x-10y&=-2\\3x-5y&=5 \end{aligned}\right.\)
    Answer

    1. \((4, −1)\)

    3. \((6, 3)\)

    5. \((x,−\frac{5}{4}x+3)\)

    Exercise \(\PageIndex{4}\) Solving Linear Systems by Substitution

    Solve by substitution.

    1. \(\left\{\begin{aligned} y&=7x−2\\x+y&=6 \end{aligned}\right.\)
    2. \(\left\{\begin{aligned} 2x−4y&=10\\x&=−2y−1 \end{aligned}\right.\)
    3. \(\left\{\begin{aligned} x−y&=0\\5x−7y&=−8 \end{aligned}\right.\)
    4. \(\left\{\begin{aligned} 9x+2y&=−41\\−x+y&=7 \end{aligned}\right.\)
    5. \(\left\{\begin{aligned} 6x−3y&=4\\2x−9y&=4 \end{aligned}\right.\)
    6. \(\left\{\begin{aligned} 8x−y&=7\\12x+3y&=6 \end{aligned}\right.\)
    7. \(\left\{\begin{aligned} 20x−4y&=−3\\−5x+y&=−12 \end{aligned}\right.\)
    8. \(\left\{\begin{aligned} 3x−y&=6\\x−13y&=2 \end{aligned}\right.\)
    9. \(\left\{\begin{aligned} x&=−1\\8x−4y&=−10 \end{aligned}\right.\)
    10. \(\left\{\begin{aligned} y&=−7\\14x−4y&=0 \end{aligned}\right.\)
    Answer

    1. \((1, 5)\)

    3. \((4, 4)\)

    5. \((\frac{1}{2}, −\frac{1}{3})\)

    7. \(Ø\)

    9. \((−1, \frac{1}{2})\)

    Exercise \(\PageIndex{5}\) Solving Linear Systems by Elimination

    Solve by elimination.

    1. \(\left\{\begin{aligned} x−y&=5\\3x−8y&=5 \end{aligned}\right.\)
    2. \(\left\{\begin{aligned} 7x+2y&=−10\\9x+4y&=−30 \end{aligned}\right.\)
    3. \(\left\{\begin{aligned} 9x−6y&=−6\\2x−5y&=17 \end{aligned}\right.\)
    4. \(\left\{\begin{aligned} 4x−2y&=30\\3x+7y&=14 \end{aligned}\right.\)
    5. \(\left\{\begin{aligned} \frac{5}{2}x−2y&=−\frac{1}{14}\\ \frac{1}{6}x−\frac{1}{3}y&=−\frac{1}{3} \end{aligned}\right.\)
    6. \(\left\{\begin{aligned} 2x−\frac{3}{2}y&=20\\ \frac{3}{32}x−\frac{1}{3}y&=\frac{1}{16} \end{aligned}\right.\)
    7. \(\left\{\begin{aligned} 0.1x−0.3y&=0.17\\0.6x+0.5y&=−0.13 \end{aligned}\right.\)
    8. \(\left\{\begin{aligned} −1.25x−0.45y&=−12.23\\0.5x−1.5y&=5.9 \end{aligned}\right.\)
    9. \(\left\{\begin{aligned} 6x−52y&=−5\\−12x+5y&=10 \end{aligned}\right.\)
    10. \(\left\{\begin{aligned} 27x+12y&=−2\\9x+4y&=3 \end{aligned}\right.\)
    11. \(\left\{\begin{aligned} 6x−5y&=0\\4x−3y&=2 \end{aligned}\right.\)
    12. \(\left\{\begin{aligned} 5x&=1\\10x+3y&=6 \end{aligned}\right.\)
    13. \(\left\{\begin{aligned} 8y&=−2x+6\\3x&=6y−18 \end{aligned}\right.\)
    14. \(\left\{\begin{aligned} 6y&=3x+1\\9x−27y−3&=0 \end{aligned}\right.\)
    Answer

    1. \((7, 2) \)

    3. \((−4, −5) \)

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

    7. \((0.2, −0.5) \)

    9. \((x,\frac{12}{5}x+2)\) 

    11. \((5, 6) \)

    13. \((−3, \frac{3}{2})\)

    Exercise \(\PageIndex{6}\) Applications of Linear Systems

    Set up a linear system and solve.

    1. The sum of two numbers is \(74\) and their difference is \(38\). Find the numbers.
    2. The sum of two numbers is \(34\). When the larger is subtracted from twice the smaller, the result is \(8\). Find the numbers.
    3. A jar full of \(40\) coins consisting of dimes and nickels has a total value of $\(2.90\). How many of each coin are in the jar?
    4. A total of $\(9,600\) was invested in two separate accounts earning \(5.5\)% and \(3.75\)% annual interest. If the total simple interest earned for the year was $\(491.25\), then how much was invested in each account? 
    5. A \(1\)% saline solution is to be mixed with a \(3\)% saline solution to produce \(6\) ounces of a \(1.8\)% saline solution. How much of each is needed?
    6. An \(80\)% fruit juice concentrate is to be mixed with water to produce \(10\) gallons of a \(20\)% fruit juice mixture. How much of each is needed?
    7. An executive traveled a total of \(4\frac{1}{2}\) hours and \(435\) miles to a conference by car and by light aircraft. Driving to the airport by car, he averaged \(50\) miles per hour. In the air, the light aircraft averaged \(120\) miles per hour. How long did it take him to drive to the airport?
    8. Flying with the wind, an airplane traveled \(1,065\) miles in \(3\) hours. On the return trip, against the wind, the airplane traveled \(915\) miles in \(3\) hours. What is the speed of the wind?
    Answer

    1. \(18\) and \(56\)

    3. \(18\) dimes and \(22\) nickels

    5. \(3.6\) ounces of the \(1\)% saline solution and \(2.4\) ounces of the \(3\)% saline solution 

    7. It took him \(1\frac{1}{2}\) hours to drive to the airport

    Exercise \(\PageIndex{7}\) Systems of Linear Inequalities (Two Variables)

    Determine whether the given point is a solution to the system of linear inequalities.

    1. \((5,-2)\); \(\left\{\begin{aligned} 5x−y&>8\\−3x+y&≤−6 \end{aligned}\right.\)
    2. \((2,3)\); \(\left\{\begin{aligned} 2x−3y&>−10\\−5x+y&>1 \end{aligned}\right.\)
    3. \((2,-10)\); \(\left\{\begin{aligned} y&<−10x\\−y&≥0 \end{aligned}\right.\)
    4. \((0,-2)\); \(\left\{\begin{aligned} y&>12x−4\\y&<−\frac{3}{4}x+2 \end{aligned}\right.\)
    Answer

    1. Yes

    3. No

    Exercise \(\PageIndex{8}\) Systems of Linear Inequalities (Two Variables)

    Graph the solution set.

    1. \(\left\{\begin{aligned} 8x+3y&≤24\\2x+3y&<12 \end{aligned}\right.\)
    2. \(\left\{\begin{aligned} x+y&≥7\\4x−y&≥0 \end{aligned}\right.\)
    3. \(\left\{\begin{aligned} x−3y&>−12\\−2x+6y&>−6 \end{aligned}\right.\)
    4. \(\left\{\begin{aligned} y&≤7\\x−y&>0 \end{aligned}\right.\)
    5. \(\left\{\begin{aligned} y&<4\\y&\geq \frac{4}{3}x+1\\y&>-x-1\end{aligned}\right.\)
    6. \(\left\{\begin{aligned} x-y&\geq -3\\x-y&\leq 3\\x+y&<1 \end{aligned}\right.\)
    Answer

    1. 

    Screenshot (452).png
    Figure 4.E.5

    3.

    Screenshot (453).png
    Figure 4.E.6

    5.

    Screenshot (454).png
    Figure 4.E.7

    Sample Exam

    Exercise \(\PageIndex{9}\)

    1. Is \((−3, 2)\) a solution to the system \(\left\{\begin{aligned}2x−3y&=−12\\−4x+y&=14\end{aligned}\right.\)?
    2. Is \((−2, 9)\) a solution to the system \(\left\{\begin{aligned}x+y&≥7\\4x−y&<0\end{aligned}\right.\)?
    Answer

    1. Yes

    Exercise \(\PageIndex{10}\)

    Given the graph, determine the simultaneous solution.

    1.

    Screenshot (455).png
    Figure 4.E.8

    2.

    Screenshot (456).png
    Figure 4.E.9
    Answer

    1. \((−1, −2)\)

    Exercise \(\PageIndex{11}\)

    Solve using the graphing method.

    1. \(\left\{\begin{aligned}y&=x−3\\y&=−12x+3 \end{aligned}\right.\)
    2. \(\left\{\begin{aligned}2x+3y&=6\\−x+6y&=−18 \end{aligned}\right.\)
    3. \(\left\{\begin{aligned}y&=2\\x+y&=3 \end{aligned}\right.\)
    4. \(\left\{\begin{aligned}y&=x\\x&=−5 \end{aligned}\right.\)
    Answer

    1. \((4, 1)\)

    3. \((1, 2)\)

    Exercise \(\PageIndex{12}\)

    Solve using the substitution method.

    1. \(\left\{\begin{aligned}5x+y&=−14\\2x−3y&=−9 \end{aligned}\right.\)
    2. \(\left\{\begin{aligned}4x−3y&=1\\x−2y&=2 \end{aligned}\right.\)
    3. \(\left\{\begin{aligned}5x+y&=1\\10x+2y&=4 \end{aligned}\right.\)
    4. \(\left\{\begin{aligned}x−2y&=4\\3x−6y&=12\end{aligned}\right.\)
    Answer

    1. \((−3, 1)\)

    3. \(Ø\)

    Exercise \(\PageIndex{13}\)

    Solve using the elimination method.

    1. \(\left\{\begin{aligned} 4x−y&=13\\−5x+2y&=−17 \end{aligned}\right.\)
    2. \(\left\{\begin{aligned} 7x−3y&=−23\\4x+5y&=7 \end{aligned}\right.\)
    3. \(\left\{\begin{aligned} −3x+18y&=1\\8x−6y&=6 \end{aligned}\right.\)
    4. \(\left\{\begin{aligned} −4x+3y&=−3\\8x−6y&=6 \end{aligned}\right.\)
    5. \(\left\{\begin{aligned} \frac{1}{2}x+\frac{3}{4}y&=\frac{7}{4}\\4x−\frac{1}{3}y&=\frac{4}{3} \end{aligned}\right.\)
    6. \(\left\{\begin{aligned} 0.2x−0.1y&=−0.24\\−0.3x+0.5y&=0.08\end{aligned}\right.\)
    Answer

    1. \((3, −1)\)

    3. \(Ø\)

    5. \((\frac{1}{2}, 2)\)

    Exercise \(\PageIndex{14}\)

    Graph the solution set.

    1. \(\left\{\begin{aligned} 3x+4y&<2\\4x−4y&<8 \end{aligned}\right.\)
    2. \(\left\{\begin{aligned} x&≤8\\3x−8y&≤0\end{aligned}\right.\)
    Answer

    1. 

    Screenshot (457).png
    Figure 4.E.10

    Exercise \(\PageIndex{15}\)

    Set up a linear system of two equations and two variables and solve it using any method.

    1. The sum of two integers is \(23\). If the larger integer is one less than twice the smaller, then find the two integers.
    2. James has $\(2,400\) saved in two separate accounts. One account earns \(3\)% annual interest and the other earns \(4\)%. If his interest for the year totals $\(88\), then how much is in each account?
    3. Mary drives \(110\) miles to her grandmother’s house in a total of \(2\) hours. On the freeway, she averages \(62\) miles per hour. In the city she averages \(34\) miles per hour. How long does she spend on the freeway?
    4. A \(15\)% acid solution is to be mixed with a \(35\)% acid solution to produce \(12\) ounces of a \(22\)% acid solution. How much of each is needed?
    5. Joey has bag full of \(52\) dimes and quarters with a total value of $\(8.35\). How many of each coin does Joey have?
    Answer

    1. \(8\) and \(15\)

    3. She drives \(1\frac{1}{2}\) hours on the freeway.

    5. \(21\) quarters and \(31\) dimes