On both lines
A solution makes both equations true at once — graphically, the intersection.
Why it works
One point fits both rules.
Linear · Systems of equations · 4.1
What is a system?
A system of two linear equations asks for the point (x, y) that satisfies BOTH — the point where the two lines cross. For y = x + 1 and y = −x + 5, that point is (2, 3).
See it work.
Watch the method
What is a system?
The solution is where the lines intersect: (2, 3).
1
The system
Two lines — the solution is the point on both.
2
Find where the lines cross
Solve for x, then back-substitute for y.
3
The solution
Write the intersection point.
Practice until it feels automatic.
Your turn
Work it out.
Type (x, y).
streak 0
best 0
…
Answer:
The idea underneath.
From easy to tricky.
Solve the system: y=2x+1 and y=x+2. Give the point (x, y).
1
The system
Two lines — the solution is the point on both. y = 2x + 1;\; y = x + 2
2
Find where the lines cross
Solve for x, then back-substitute for y. x = 1
3
The solution
Write the intersection point. (1, 3)
Solve the system: y=x+4 and y=2x+5. Give the point (x, y).
1
The system
Two lines — the solution is the point on both. y = x + 4;\; y = 2x + 5
2
Find where the lines cross
Solve for x, then back-substitute for y. x = -1
3
The solution
Write the intersection point. (-1, 3)
Solve the system: y=-x+5 and y=-2x+7. Give the point (x, y).
1
The system
Two lines — the solution is the point on both. y = -x + 5;\; y = -2x + 7
2
Find where the lines cross
Solve for x, then back-substitute for y. x = 2
3
The solution
Write the intersection point. (2, 3)
Where students slip.
Trap
Giving only x.
Fix
A system solution is a point (x, y).
Trap
Checking just one equation.
Fix
It must satisfy BOTH.
Test yourself.
The Test
12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.
0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so
t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).Core
Core (4)
C1
Solve the system: y=-x+1 and y=-2x-2. Give the point (x, y).
Need a hint?
- Find where the lines cross.
- Give the point (x, y).
C2
Solve the system: y=-x+5 and y=2x-7. Give the point (x, y).
Need a hint?
- Find where the lines cross.
- Give the point (x, y).
C3
Solve the system: y=x-5 and y=3x-9. Give the point (x, y).
Need a hint?
- Find where the lines cross.
- Give the point (x, y).
C4
Solve the system: y=2x-1 and y=3x-4. Give the point (x, y).
Need a hint?
- Find where the lines cross.
- Give the point (x, y).
Apply
Apply (4)
A1
Solve the system: y=-2x+7 and y=2x-5. Give the point (x, y).
Need a hint?
- Find where the lines cross.
- Give the point (x, y).
A2
Solve the system: y=-x+8 and y=-2x+11. Give the point (x, y).
Need a hint?
- Find where the lines cross.
- Give the point (x, y).
A3
Solve the system: y=-x+9 and y=2x-3. Give the point (x, y).
Need a hint?
- Find where the lines cross.
- Give the point (x, y).
A4
Solve the system: y=x-5 and y=-2x+1. Give the point (x, y).
Need a hint?
- Find where the lines cross.
- Give the point (x, y).
Extension
Extension (4)
E1
Solve the system: y=x-7 and y=-2x+5. Give the point (x, y).
Need a hint?
- Find where the lines cross.
- Give the point (x, y).
E2
Solve the system: y=-x+5 and y=3x-7. Give the point (x, y).
Need a hint?
- Find where the lines cross.
- Give the point (x, y).
E3
Solve the system: y=-2x-5 and y=3x+10. Give the point (x, y).
Need a hint?
- Find where the lines cross.
- Give the point (x, y).
E4
Solve the system: y=2x+8 and y=x+5. Give the point (x, y).
Need a hint?
- Find where the lines cross.
- Give the point (x, y).