4.E: Review Exercises and Sample Exam

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.

Figure 4.E.1

2.

Figure 4.E.2

3.

Figure 4.E.3

4.

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.

Figure 4.E.5

3.

Figure 4.E.6

5.

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.

Figure 4.E.8

2.

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.

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