Introduction to Simultaneous Equations
Discover how two linear equations can be solved at the same time by finding where their graphs intersect.
What are Simultaneous Equations?
Simultaneous equations are two (or more) equations that are true at the same time. The solution is a pair of values (x, y) that satisfies both equations simultaneously.
A Real-World Example
Amara and Ben are saving money. Amara has $20 and saves $5 per week. Ben has $4 and saves $7 per week.
Amara: y = 5x + 20
Ben: y = 7x + 4
After how many weeks will they have the same amount? This is where the two lines intersect!
The Graphical Method
The solution is the point of intersection — the x and y values where both lines meet.
The two lines meet at (2, 3) — so x = 2 and y = 3 is the solution to both equations.
Steps for the Graphical Method
-
1
Graph both equations on the same set of axes
Use a table of values or gradient-intercept method for each line.
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2
Identify the point of intersection
Read the coordinates (x, y) where the two lines cross.
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3
Check your solution in both equations
Substitute x and y back into both original equations to verify they are correct.
When do lines NOT intersect?
Parallel lines (same gradient, different y-intercept) never intersect — so there is no solution. Identical lines have infinitely many solutions.
Creating a Table of Values
To graph a line, substitute x values to find corresponding y values.
y = 2x − 1
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | −1 | 1 | 3 | 5 |
y = x + 1
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 1 | 2 | 3 | 4 |
At x = 2: both give y = 3. So the solution is (2, 3).
Key Vocabulary
Simultaneous Equations
Two or more equations that must be satisfied by the same values of the variables at the same time.
Point of Intersection
The point where two lines cross. Its coordinates give the solution to the simultaneous equations.
Solution
The pair (x, y) that makes both equations true at the same time.
Parallel Lines
Lines with the same gradient that never meet — representing equations with no solution.
Worked Examples
Solve graphically: y = x + 2 and y = 3x − 2
Step 1: Table for y = x + 2: (0,2), (1,3), (2,4)
Step 2: Table for y = 3x − 2: (0,−2), (1,1), (2,4)
Step 3: Both lines give y = 4 when x = 2 → intersection at (2, 4)
Check: y = 2 + 2 = 4 ✓ and y = 3(2) − 2 = 4 ✓
Answer: x = 2, y = 4
Do y = 2x + 1 and y = 2x − 3 have a solution?
Step 1: Both lines have gradient m = 2 — they are parallel.
Step 2: Parallel lines never intersect.
Answer: No solution — the lines are parallel.
Check if (1, 5) is the solution to y = 3x + 2 and y = −x + 6.
Equation 1: y = 3(1) + 2 = 5 ✓
Equation 2: y = −(1) + 6 = 5 ✓
Answer: Yes, (1, 5) is the solution — it satisfies both equations.
Knowledge Check
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Key Concepts Summary
- ●Simultaneous equations have the same solution (x, y) that satisfies both equations.
- ●Graphically, the solution is the point of intersection of the two lines.
- ●Always check your solution by substituting back into both original equations.
- ●Parallel lines (same gradient) have no solution.
- ●Use a table of values to help sketch each line accurately.