Download Math 006: Systems of Linear Equations Hints and Answers 1. Solve

Math 006: Systems of Linear Equations
Hints and Answers
1. Solve
y = 5 − 2x
x + 2y = 4
by graphing.
Answer: The lines graphed below intersect at (2, 1), so the solution is (2, 1).
2. (a) What type of linear system has equations whose graphs are two parallel but nonintersecting lines?
(b) What type of linear system has equations whose graphs are two parallel lines that intersect?
(c) What type of linear system has equations whose graphs are two nonparallel lines?
Answer: (a) Inconsistent
(b) Dependent
(c) Independent
3. Solve
x − 3y = 3
2x − 6y = 12
by substitution.
Answer: The first equation implies x = 3 + 3y. Substitute the quantity 3 + 3y for x in the
second equation to obtain 2(3 + 3y) − 6y = 12, or 6 + 6y − 6y = 12, or 6 = 12. This is never
true, so the system is inconsistent and has no solution.
4. Solve
7x − y = 4
5x + 2y = 1
by substitution.
Answer: Solve the first equation for y to obtain y = 7x − 4. Substitute the quantity 7x − 4 for
y in the second equation to obtain 5x + 2(7x − 4) = 1. Then 5x + 14x − 8 = 1, or 19x − 8 = 1
63
and thus 19x = 9. Then x = 9/19,
and y = 7(9/19) − 4. That is, y = 19
− 76
= − 13
. The
19
19
9
13
solution to the system is 19 , − 19 .
5. Solve
4x − 8y = 36
3x − 6y = 27
using the addition method.
Answer: Multiply the first equation by 3 and the second by −4 and then add them together:
12x − 24y =
108
−12x + 24y = −108
0 =
0
This is always true, so the system is dependent.
6. Solve
4x − 2y = 5
2x + 3y = 4
using the addition method.
Answer: Multiply the second equation by −2 and then add the equations (this will eliminate
x):
4x −
−4x −
2y =
5
6y = −8
−8y = −3
−3
= 38 . Substituting this value for y in the first equation yields 4x −
Thus y = −8
23 3
4x = 23
and so x = 23
. The solution to the system is 16
,8 .
4
16
3
4
= 5. Then
7. A rowing team rowing with the current traveled 40 km in 2 h. Against the current, the
team could only travel 16 km in 2 h. Find the team’s rowing rate in calm water and the rate
of the current.
Answer: Let r be the rate of the rowing team in calm water and let c be the rate of the
current.
Direction
Rate Time
Distance
With current
r+c
2
2(r + c) = 40
Against current r − c
2
2(r − c) = 16
2r + 2c = 40
. Adding the two equations together pro2r − 2c = 16
vides 4r = 56 and so r = 14, and then c = 6. Thus the rate of the rowing team in calm water
is 16 km/hr and the rate of the current is 6 km/hr.
Therefore, we solve the system
Further Practice Related to Khan Academy Videos
8. Solve the system
y = 3x + 1
by graphing
2y + 4 = 6x
Answer: See the Khan Academy Video Algebra I: Solving systems by graphing 3
9. Solve the following systems of equations using the addition method:
(a)
9x − 4y = −78
4x + y = −18
(b)
−7x − 4y = 9
x + 2y = 3
(c)
−3x − 9y =
66
−7x + 4y = −71
(d)
3t + 5d = 62.5
3t + 2d =
52
(e)
3t + 4g = 6
−6t + g = 6
(f)
2x − y =
14
−6x + 3y = −42
(g) The sum of two numbers is 70. Their difference is 24. What are the two numbers?
(h)
4x − 2y =
5
2x − y = 2.5
(check your solution to (h) by graphing)
Answer: For (a), (b) and (c), see the Khan Academy Video Algebra: Systems of Equations
(d) See the Khan Academy Video Algebra I: Solving systems by elimination
(e) See the Khan Academy Video Algebra I: Solving systems by elimination 2
(f) See the Khan Academy Video Developmental Math 2: Addition Elimination Method 3
(g) See the Khan Academy Video Developmental Math 2: Addition Elimination Method 4
(h) See the Khan Academy Video Algebra I: Solving systems by elimination 3
10. Solve the following systems of equations by substitution:
(a)
2y = x + 7
x = y − 4
(b)
y = −5x + 8
10x + 2y = −2
(c)
y = − 41 x + 100
y = − 14 x + 120
(d)
y = 4x − 17.5
y + 2x = 6.5
(e)
9x + 3y = 15
y − x = 5
Check your answer to (e) by graphing.
(f) A coin bank contains total of 16 coins consisting of nickels and quarters. The total value of
coins in the bank is $2.00. Find how many nickels and how many quarters are in the bank. Do
this by creating and solving a linear system where n is the number nickels and q is the number
of quarters.
Answer: (a) See the Khan Academy Video Developmental Math 2: Substitution Method
(b) See the Khan Academy Video Developmental Math 2: Substitution Method 2
(c) See the Khan Academy Video Algebra I: Solving systems by substitution
(d) See the Khan Academy Video Algebra I: Solving systems by substitution 2
(e) See the Khan Academy Video Algebra I: Solving systems by substitution 3
(f) See the Khan Academy Video Developmental Math 2: Substitution Method 3
Some further examples from Khan Academy on solving systems of equations by graphing
can be seen in the videos:
Algebra I: Solving systems by graphing and Algebra I: Solving systems by graphing 2