Systems of Equations & Inequalities — Practice
Two conditions at once, solved three ways — graph, substitute, or eliminate. The shaded overlap is the answer for inequalities.
What these questions cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Inequality — A statement that one expression is less than or greater than another.
- Perimeter — Distance all the way around the outside.
- Volume — Space inside, in cubic units.
- System — Two or more equations that must hold at the same time.
- Parallel — Same slope, different intercept.
- Boundary line — The line you would get by replacing the inequality sign with =.
Problem 1
Solve y = 3x - 1 and y = x + 5.
Worked answer
(3, 8)
3x - 1 = x + 5
2x = 6, so x = 3
y = 8
Problem 2
Solve x + y = 12 and x - y = 2.
Worked answer
(7, 5)
2x = 14, so x = 7
y = 5
Problem 3
Solve 3x + y = 14 and 2x - y = 1.
Worked answer
(3, 5)
5x = 15, so x = 3
9 + y = 14, so y = 5
Problem 4
Is (4, -1) a solution of 2x + 3y = 5 and x - y = 5? Answer yes or no.
Worked answer
yes
8 - 3 = 5 ✓
4 - (-1) = 5 ✓
Problem 5
For y > 2x + 1, is the boundary line solid or dashed?
Worked answer
dashed
> does not include equality
So the line is dashed
Problem 6
Solve 2x + 3y = 13 and 5x - y = 7.
Worked answer
(2, 3)
15x - 3y = 21
Add: 17x = 34, so x = 2
10 - y = 7, so y = 3
Problem 7
Solve 4x + 5y = 7 and 3x - 2y = 12.
Worked answer
(3, -1)
× 2: 8x + 10y = 14; × 5: 15x - 10y = 60
Add: 23x = 69, so x = 3
12 + 5y = 7, so y = -1
Problem 8
For y ≤ -x + 6, does the shading include (0, 0)? Answer yes or no.
Worked answer
yes
0 ≤ 6 is true
So the origin is in the solution region
Problem 9
How many solutions does 2x - y = 4 and 4x - 2y = 9 have?
Worked answer
none
Double the first: 4x - 2y = 8
But the second says 9
Parallel lines, so no solution
Problem 10
Two numbers add to 54; one is twice the other. Find the larger.
Worked answer
36
2y + y = 54
y = 18
Larger is 36
Problem 11
5 pens and 3 pencils cost $4.10. 2 pens and 3 pencils cost $2.30. Find the price of one pen.
Worked answer
0.6
3p = 1.80
p = $0.60
Problem 12
A system of two linear inequalities has a solution region shaped like a wedge. This means
- the two boundary lines cross
- the lines are parallel
- there is no solution
- the lines are the same
Worked answer
the two boundary lines cross
Crossing lines cut the plane into four wedges
The solution is one of them
Problem 13
A 30% acid solution is mixed with a 60% solution to make 10 litres of 50% solution. How many litres of the 30% solution are used?
Worked answer
10/3
x + y = 10
0.3x + 0.6y = 5
0.3x + 0.6(10 - x) = 5
-0.3x = -1, so x = 103 ≈ 3.33 L
Problem 14
Solve 6x - 4y = 10 and 9x - 6y = 15.
Worked answer
infinitely many
First ÷ 2: 3x - 2y = 5
Second ÷ 3: 3x - 2y = 5
The same line, so infinitely many solutions
Problem 15
A rectangle's perimeter is 44 cm and its length is 3 cm more than twice its width. Find the width.
Worked answer
19/3
2(2w + 3) + 2w = 44
6w + 6 = 44
6w = 38, so w = 193 ≈ 6.33 cm
Problem 16
A point (3, 5) satisfies y > x + 1 and y ≤ 2x. Verify it. Answer yes or no.
Worked answer
yes
5 > 4 ✓
5 ≤ 6 ✓
So yes, it is in the solution region