Systems of Equations & Inequalities — Challenge
Two conditions at once, solved three ways — graph, substitute, or eliminate. The shaded overlap is the answer for inequalities.
What these problems 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
If x+y=7 and x-y=3, what is x?
- 2
- 3
- 4
- 5
- 10
Worked answer
5
Add the two equations so that y cancels: 2x = 10.
So x = 5, and then y = 2.
Check: 5+2 = 7 and 5-2 = 3.
Problem 2
If 2x+3y=12 and x=3, what is y?
- 1
- 2
- 3
- 4
- 6
Worked answer
2
Substitute x=3: 2(3)+3y = 12, so 6+3y = 12.
Subtract 6: 3y = 6, so y = 2.
Problem 3
If 3x-y=5 and x+y=7, what is y?
- 2
- 3
- 4
- 5
- 7
Worked answer
4
Add the equations to eliminate y: 4x = 12, so x = 3.
Substitute into x+y = 7: 3+y = 7, so y = 4.
Check the first equation: 3(3)-4 = 5.
Problem 4
Where do the lines y=x+2 and y=2x-1 cross?
- (1,3)
- (3,5)
- (2,4)
- (5,3)
- (0,2)
Worked answer
(3,5)
Set the two expressions for y equal: x+2 = 2x-1.
Subtract x and add 1: x = 3.
Then y = 3+2 = 5, so the crossing point is (3,5).
Problem 5
If x=2y and x+y=12, what is y?
- 3
- 4
- 6
- 8
- 12
Worked answer
4
Substitute x = 2y into the second equation: 2y+y = 12.
So 3y = 12 and y = 4.
Then x = 8, and 8+4 = 12 as required.
Problem 6
If 4x+2y=10 and y=1, what is x?
- 1
- 2
- 3
- 4
- 8
Worked answer
2
Substitute y=1: 4x+2 = 10.
Subtract 2: 4x = 8, so x = 2.
Problem 7
Three adult tickets and two child tickets cost $32. An adult ticket costs $8. What does a child ticket cost in dollars?
- $2
- $3
- $4
- $5
- $8
Worked answer
$4
Three adult tickets cost 3· 8 = 24 dollars.
That leaves 32-24 = 8 dollars for two child tickets.
So one child ticket costs 82 = 4 dollars.
Problem 8
A system of two linear equations has infinitely many solutions. What does that mean about the lines?
- They are parallel and distinct
- They are the same line
- They cross at one point
- There are no lines
- They are perpendicular
Worked answer
They are the same line
A solution is a point lying on both lines, so infinitely many solutions means infinitely many shared points.
Two distinct straight lines can share at most one point, so the two equations must describe the same line.
Parallel distinct lines share no points at all, giving no solution instead.