Systems of Linear Equations — Drills
Systems of Linear Equations has 8 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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.
Substitution
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve y = 4x - 4 and 2x + y = 32.
Worked answer
(6, 20)
2x + 4x - 4 = 32
6x = 36
x = 6
y = 20
Problem 2
Solve y = 3x + 5 and 1x + y = -11.
Worked answer
(-4, -7)
1x + 3x + 5 = -11
4x = -16
x = -4
y = -7
Elimination
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve 4x + 6y = -4 and 4x - 6y = -28.
Worked answer
(-4, 2)
Add: 8x = -32, so x = -4
-16 + 6y = -4
y = 2
Problem 2
Solve 1x + 5y = -5 and 1x - 5y = 5.
Worked answer
(0, -1)
Add: 2x = 0, so x = 0
0 + 5y = -5
y = -1
Checking a solution
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Is (1, 3) a solution of 5x + 1y = 8 and x - y = -2? Answer yes or no.
Worked answer
yes
First: 5(1) + 1(3) = 8, and we need 8.
Second: 1 - (3) = -2, and we need -2.
Both check out, so yes.
Problem 2
Is (8, 1) a solution of 3x + 5y = 20 and x - y = 4? Answer yes or no.
Worked answer
no
First: 3(8) + 5(1) = 29, and we need 20.
Second: 8 - (1) = 7, and we need 4.
At least one fails, so no.
No solution
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
How many solutions does y = 7x + 0 and y = 7x + 0 have?
- Exactly one
- Infinitely many
- Exactly two
- None
Worked answer
Infinitely many
Both have slope 7.
Same intercept too, so it is the same line.
Infinitely many solutions.
Problem 2
How many solutions does y = 2x - 3 and y = 2x - 2 have?
- None
- Exactly one
- Infinitely many
- Exactly two
Worked answer
None
Both have slope 2.
Different intercepts, so the lines are parallel.
They never meet, so no solution.
From a graph
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two lines cross at (2, 4). What is the solution of the system?
Worked answer
(2, 4)
The crossing point is the one pair satisfying both equations.
(2, 4)
Problem 2
Two lines cross at (6, 1). What is the solution of the system?
Worked answer
(6, 1)
The crossing point is the one pair satisfying both equations.
(6, 1)
Word problems
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two numbers add to 63 and differ by 11. Find the larger.
Worked answer
37
x + y = 63 and x - y = 11
Add: 2x = 74
x = 37
Problem 2
Two numbers add to 24 and differ by 6. Find the larger.
Worked answer
15
x + y = 24 and x - y = 6
Add: 2x = 30
x = 15
Multiplying
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve x + 3y = 8 and 2x - y = 2.
Worked answer
(2, 2)
× 3: 6x - 3y = 6
Add to the first: 7x = 14
x = 2, then y = 2
Problem 2
Solve x + 2y = 5 and 4x - y = 20.
Worked answer
(5, 0)
× 2: 8x - 2y = 40
Add to the first: 9x = 45
x = 5, then y = 0
Infinite
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
How many solutions does 9x + 9y = 36 and 3x + 3y = 12 have?
Worked answer
infinitely many
Divide the first by 3: 3x + 3y = 12
Identical to the second.
The same line, so infinitely many solutions.
Problem 2
How many solutions does 8x + 6y = 12 and 4x + 3y = 6 have?
Worked answer
infinitely many
Divide the first by 2: 4x + 3y = 6
Identical to the second.
The same line, so infinitely many solutions.