4.E: Review Exercises and Sample Exam

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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.

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.

3.

4.

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.

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.

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.

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. 3. 5. 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. 2. 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. 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?
1. $$8$$ and $$15$$
3. She drives $$1\frac{1}{2}$$ hours on the freeway.
5. $$21$$ quarters and $$31$$ dimes