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MTH-4101-2 C ANSWER KEY
Equations & Inequalities II
1. What is the solution of the following system of equations? Write your answer as
an ordered pair.
y
(–1.5, 2)
x
/5
2. Determine the solution set for the following system of equations:
y – 4 = 3x – 1
–3
x
–6
y
–2
–3
–1
0
0
3
y–x=1
x
y
–1
0
0
1
1
2
–2
–1
/3
3. Without solving, express the following as a system of two equations, making sure
to define your variables.
If I buy 2 shirts and 3 ties, my bill comes to $213. However, if I buy 5 ties and 3
shirts my bill comes to $337. What is the price of a single shirt and tie?
Let x = price of 1 shirt
Let y = price of 1 tie
1) 2x + 3y = 213
2) 5x + 3y = 337
/5
4. Solve the following system of equations graphically. Write your solution as an
ordered pair.
1) 4y = 2 – 5x
2) 6 – 2y = 0
Ans: (–2, 3)
y
x
/5
5. Solve the following system of equations graphically. Write your solution as an
ordered pair.
1) 2y + x = 0
2)
3
/2x = 3 – y
Ans: (3, –1.5)
y
x
/5
6. Using the method of your choice, determine the solution of the following system
of equations; write your answer as an ordered pair:
1)
2)
1
2
/2x – 1/3y = 1
/3x + 1/2y = –2
1) 1/2x – 1/3y = 1
→ 1) 1/2x = 1/3y + 1
2) 2/3x + 1/2y = –2
→
→
2) 2/3x = –1/3y – 2
1) x = 2(1/3y + 1)
→
→ 2) x = 3/2(–1/3y – 2)
1) x = 2/3y + 2
2) x = –1/2y – 3
→
x=x
/3y + 2 = –1/2y – 3
2
/3y + 1/2y = – 3 – 2
7
/6y = –5
y = 6/7(–5)
2
y = –30/7
→
1) x = 2/3(–30/7) + 2
x = –20/7 + 2
x = –6/7
ANS: (–6/7, –30/7)
/10
7. Using the method of your choice, determine the solution of the following system
of equations; write your answer as an ordered pair:
1) x = 5y – 15
2) y = 1/5x – 1
x = 5(1/5x – 1) – 15
x = x – 5 – 15
x – x = –20
0x = –20
ANS: Parallel lines, no solution
/10
8. Using the method of your choice, determine the solution of the following system
of equations; write your answer as an ordered pair:
1) 5x + 2y = 4x + 8
2) 4y = 20 – 3x
1) 5x + 2y = 4x + 8
2) 4y = 20 – 3x
→ 1) 5x – 4x = –2y + 8
→ 2) 3x = –4y + 20
→ 1) x = –2y + 8
→ 2) x = 1/3(–4y + 20)
→
2) x = –4/3y + 20/3
x=x
/3y + 20/3 = –2y + 8
–4
/3y + 2y = 8 – 20/3
2
/3y = 4/3
y = 3/2(4/3)
–4
y=2
→ 1) x = –2(2) + 8
x=4
ANS: (4, 2)
/10
9. During a Montreal Alouettes football game, I buy 2 hamburgers and 3 hotdogs
and it costs me $8.25. The next game, I buy 4 hamburgers and 5 hotdogs for
$15.25. What is the price of 1 hotdog and 1 hamburger?
Let x = price of 1 hotdog
Let y = price of 1 hamburger
1) 3x + 2y = 8.25
2) 5x + 4y = 15.25
1) 3x + 2y = 8.25
2) 5x + 4y = 15.25
→
→
3x = 8.25 – 2y → x = 8.25/3 – 2/3y
5x = 15.25 – 4y → x = 15.25/5 – 4/5y
– 2/3y = 15.25/5 – 4/5y
–2/ y + 4/ y = 15.25/ – 8.25/
3
5
5
3
2/ y = 3/
15
10
8.25/
3
y = 2.25
→
x = 8.25/3 – 2/3(2.25)
x = 2.75 – 1.5
x = 1.25
ANS: Hotdogs cost $1.25 each, Hamburgers cost $2.25 each
/9
10. The difference between two numbers is 9. One-sixth the smaller number added to
one-third the larger number is equal to 12. What are the two numbers?
Let x = smaller number
Let y = large number
1) y – x = 9
2) 1/6x + 1/3y = 12
1) y – x = 9 →
2) 1/6x + 1/3y = 12
y=9+x
1/ y = 12 – 1/ x →
→
3
6
y = 3(12 – 1/6x) →
y = 36 – 1/2x
y=y
9 + x = 36 – 1/2x
9 – 36 = –1/2x – x
–27 = –3/2x
–2
–27( /3) = x
x = 18
→
1) y = 9 + (18)
y = 27
ANS: the smaller # is 18, the larger # is 27
/9
11. A brother is 5 years older than his sister. Three years from now, the sister will be
three-quarters of her brother’s age then. How old are the brother and sister now?
Let x = brother’s age
Let y = sister’s age
1) x = y + 5
2) y + 3 = ¾(x + 3)
2) y + 3 = ¾(x + 3) → y + 3 = ¾x + 9/4 →
y + 3 – 9/4 = ¾x →
4/ (y
3
+ 3/4) = x
→
x=x
y + 5 = 4 /3 y + 1
y – 4 /3 y = 1 – 5
–1/ y = –4
3
y = –3(–4)
y = 12
→
x = (12) + 5
x = 17
ANS: the brother is 17 years old, the sister is 12 years old
/9
12. Given the adjacent system of inequalities, state whether or not the following
points belong to the solution set:
y
a) (1, 2)
NO
b) (2½, 3/4) YES
x
/4
4/ y
3
+1=x
13. Solve the following system of inequalities graphically:
1) x + 2y ≤ 4
2) 3x – y ≥ 3
y
x
ANS
/8
14. Solve the following system of inequalities graphically:
1) x + y < 2
2) x ≥ y + 1
y
x
ANS
/8