Systems of Linear Equations — Practice
Two conditions at once. The answer is the single pair of numbers that satisfies both.
What these questions cover
These questions use 3 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
Problem 1
Solve y = 2x and x + y = 9.
Worked answer
(3, 6)
x + 2x = 9
3x = 9, so x = 3
y = 6
Problem 2
Solve y = x + 4 and y = 3x.
Worked answer
(2, 6)
x + 4 = 3x
4 = 2x, so x = 2
y = 6
Problem 3
Solve x + y = 10 and x - y = 4.
Worked answer
(7, 3)
2x = 14, so x = 7
7 + y = 10, so y = 3
Problem 4
Solve 2x + y = 11 and 2x - y = 5.
Worked answer
(4, 3)
Add: 4x = 16, so x = 4
8 + y = 11, so y = 3
Problem 5
Is (2, 5) a solution of y = 3x - 1 and x + y = 7? Answer yes or no.
Worked answer
yes
3(2) - 1 = 5 ✓
2 + 5 = 7 ✓
Yes
Problem 6
Solve y = 4x - 3 and 2x + y = 15.
Worked answer
(3, 9)
2x + 4x - 3 = 15
6x = 18, so x = 3
y = 9
Problem 7
Solve 3x + 2y = 16 and x - 2y = 0.
Worked answer
(4, 2)
4x = 16, so x = 4
4 - 2y = 0, so y = 2
Problem 8
Solve 5x + 3y = 21 and 5x - y = 9.
Worked answer
(2.4, 3)
4y = 12, so y = 3
5x - 3 = 9, so 5x = 12
x = 2.4
Problem 9
How many solutions does y = 3x + 1 and y = 3x - 4 have?
- None
- Exactly one
- Infinitely many
- Exactly two
Worked answer
None
Same slope 3, different intercepts
The lines are parallel
They never meet
Problem 10
Two lines cross at (5, -2). What is the solution of the system?
Worked answer
(5, -2)
The crossing point satisfies both equations
(5, -2)
Problem 11
Two numbers add to 30 and differ by 8. Find them.
Worked answer
19 and 11
Add: 2x = 38, so x = 19
y = 11
Check: 19 + 11 = 30 and 19 - 11 = 8 ✓
Problem 12
Solve x + 3y = 14 and 2x - y = 7.
Worked answer
(5, 3)
2x + 6y = 28
Subtract: 7y = 21, so y = 3
x + 9 = 14, so x = 5
Problem 13
Adult tickets cost $12 and child tickets $7. 40 tickets sold for $355. How many adult tickets?
Worked answer
15
a + c = 40 and 12a + 7c = 355
c = 40 - a, so 12a + 280 - 7a = 355
5a = 75, so a = 15
Problem 14
A boat travels 36 km downstream in 2 hours and 36 km upstream in 3 hours. Find the speed of the boat in still water.
Worked answer
15
b + c = 18 and b - c = 12
Add: 2b = 30
b = 15 km/h
Problem 15
How many solutions does 2x + 4y = 10 and x + 2y = 5 have?
Worked answer
infinitely many
Divide the first by 2: x + 2y = 5
Identical to the second
The same line, so infinitely many solutions
Problem 16
Solve 4x + 3y = 10 and 3x - 2y = 16.
Worked answer
(4, -2)
× 2: 8x + 6y = 20; × 3: 9x - 6y = 48
Add: 17x = 68, so x = 4
16 + 3y = 10, so y = -2