Math 2814 – Module 3 Sample Questions
Please complete the questions first on your own. Use these solutions only to double check your
answers. If you have trouble solving any of the questions on your own, send me email or post your
problem instead of simply looking up the answers.
Note: These questions cover CLO 2, 3, 4, and 7.
Chapter 3.1
1. Which of the following equations are linear?
3
a. 𝑥 − 𝜋𝑦 + √2 𝑧 = 0
Answer:
Yes, linear
3
b. 𝑥 − 𝜋𝑦 + √2𝑧 = 0
Answer
No, not linear
c. 𝑥 2 + 𝑦 2 + 𝑧 2 = 1
Answer
No, not linear
d. 𝑥 −1 + 7𝑦 + 𝑧 = 5
Answer
No, not linear
𝜋
e. 𝑥 + 7𝑦 + 𝑧 = sin( 9 )
Answer
Yes, linear
2. Determine graphically whether each system has a unique solution, infinitely many solutions, or
no solution. Then solve each system to confirm your answer
𝑥+𝑦 =0
a.
2𝑥 + 𝑦 = 3
Answer
x=3, y=-3
b.
𝑥 − 2𝑦 = 7
3𝑥 + 𝑦 = 7
c.
3𝑥 − 6𝑦 = 3
−𝑥 + 2𝑦 = 1
Answer
x=3, y=-2
Answer
No solution
3. Solve each of the following systems by back substitution
𝑥+𝑦 =1
a.
𝑦=3
Answer
y=3,x=-2
𝑥−𝑦+𝑧 =0
b. 2𝑦 − 𝑧 = 1
3𝑧 = −1
Answer
z=-1/3, y=1/3, x=2/3
𝑥1 + 𝑥2 − 𝑥3 − 𝑥4 = 1
𝑥2 + 𝑥3 + 𝑥4 = 0
c.
𝑥3 − 𝑥4 = 0
𝑥4 = 1
Answer
x4 = 1, x3 = 1, x2 = -2, x1 = 5
4. Find the augmented matrix for the following systems
𝑥−𝑦 =0
a.
2𝑥 + 𝑦 = 3
Answer
1
𝐴=(
2
𝑥 + 5𝑦 = −1
b. 2𝑥 + 4𝑦 = 4
−𝑥 + 𝑦 = −5
Answer
−1 0
)
1 3
1 5 −1
𝐴=( 2 4 4 )
−1 1 −5
c.
2𝑥1 + 3𝑥2 − 𝑥3 = 1
𝑥1 +
𝑥3 = 0
−𝑥1 + 2𝑥2 − 2𝑥3 = 0
Answer
2 3 −1 1
𝐴 = ( 1 0 1 0)
−1 2 −2 0
d.
𝑎 − 2𝑏 +
𝑑=2
−𝑎 + 𝑏 − 𝑐 − 3𝑑 = 1
Answer
1 −2 0
1 2
𝐴=(
)
−1 1 −1 −3 1
5. Find the linear system of equations with the given augmented matrix
0 1 1 1
a. [1 −1 0| 1]
2 −1 1 1
Answer
𝑥2 + 𝑥3 = 1
𝑥1 − 𝑥2
=1
2𝑥1 − 𝑥2 + 𝑥3 = 1
1 −1 0 1
2
b. [1 1 2 −1| 4]
0 1 0 3
0
Answer
𝑥1 − 𝑥2 + 𝑥4 = 2
𝑥1 + 𝑥2 + 2𝑥3 − 𝑥4 = 4
𝑥2 + 3𝑥4 = 0
6. Solve the systems in 4 (a) through (d) and 5 (a) and (b)
Answer
4a: x=1, y=1
4b: x1=2/3, x2=-1/3, x3=-2/3
4c: x=4, y=-1
4d: a = −4 − 2s − 5t, b = −3 − s − 2t, c=s, d=t
5a: no solution
7. Convert each nonlinear system into a linear one by substitution Then solve it
a.
2
𝑥
3
𝑥
3
𝑦
4
+𝑦
+ =0
=1
Answer
u=3, v=−2, or x=1/u=1/3, y=1/v=− 1/2
b.
𝑥 2 + 2𝑦 2 = 6
𝑥2 − 𝑦2 = 3
Answer
u=x^2=4, v=y^2=1 so x=+/-2 and y = +/-1
Section 3.2
1. Use the elementary row operation to reduce the given matrix to (a) row echelon form and (b)
reduced row echelon form
0 0 1
a. [0 1 1]
1 1 1
Answer
Swap row 3 with row 1 for row echelon form
Transform into identity matrix for reduced row echelon form
4
b. [
2
3
]
1
Answer
1
1
1 2
Row echelon form: [
], Reduced form [
0
0 1
3 −2 −1
c. [2 −1 −1]
4 −3 −1
Answer
0
]
1
1 0
Row Echelon: [0 1
0 0
−1
1
−1], reduced form [0
0
0
2 0
0 1]
0 0
2. Show that the following matrices are row equivalent
1 2
3 −1
a. 𝐴 = [
] and 𝐵 = [
]
3 4
1 0
Answer
Do the following to A: R2=R2 − 2R1, R1=−1/2R1, and R1=R1 + 7/2R2 gives matrix B
2 0 −1
3
b. 𝐴 = [ 1 1 0 ] and 𝐵 = [3
−1 1 1
2
1 −1
5 1]
2 0
Answer
Do the following to A: R1=R1+R2, R2=R2+2R3, R3=R3+R1, R2=R2+R1, R2=R2+ 1/2R3 gives matrix B
3. What is the rank of the following matrices
7 0 1 0
a. [0 1 −1 4]
0 0 0 0
Answer
Rank is the number of non-zero rows in the row echelon form of a matrix. So Rank = 2
0 0 1
b. [0 1 0]
1 0 0
Answer
Rank = 3
1
1
c. [
0
0
2
0
1
0
3
0
]
1
1
Answer
Rank = 3
4. Solve the given system either by Gaussian or Gauss-Jordan elimination
𝑥1 + 2𝑥2 − 3𝑥3 = 9
a. 2𝑥1 − 𝑥2 + 𝑥3 = 0
4𝑥1 − 𝑥2 + 𝑥3 = 4
Answer
X1=2, x2=5, x3=1
b.
2𝑟 + 𝑠 = 3
4𝑟 + 𝑠 = 7
2𝑟 + 5𝑠 = −1
Answer
r=2, s=-1
−𝑥1 + 3𝑥2 − 2𝑥3 + 4𝑥4 = 0
c. 2𝑥1 − 6𝑥2 + 𝑥3 − 2𝑥4 = −3
𝑥1 − 3𝑥2 + 4𝑥3 − 8𝑥4 = 2
Answer
x1=3s–2, x2=s, x3=2t+1, x4=t
d.
𝑤 + 𝑥 + 2𝑦 + 𝑧 = 1
𝑤−𝑥−𝑦+𝑧 =0
𝑥 + 𝑦 = −1
𝑤+𝑥+𝑧 =2
Answer
No solution
5. Find the line of intersection of the planes 3𝑥 + 2𝑦 + 𝑧 = −1 and 2𝑥 − 𝑦 + 4𝑧 = 5
Answer
z will be a free variable. So, let z = 7t. Then y=10t-17/7 and x=-9t+9/7. But that’s the parametric equation
of a line, the line of intersection!
6. Prove that if 𝑎𝑑 − 𝑏𝑐 ≠ 0, then the system
𝑎𝑥 + 𝑏𝑦 = 𝑟
has a unique solution.
𝑐𝑥 + 𝑑𝑦 = 𝑠
7. Give an example of three planes with a common line of intersection
Answer
8. Give an example of three planes that intersect in pairs but have no common point of
intersection amongst all three planes
Answer
9. Give an example of three planes that intersect in a single point
Answer