Grade 8 · Grade 8: Linear Equations, Systems and Functions · Linear Relationships and Functions
Systems of Linear Equations
Find where two lines meet, graphically and algebraically, and use systems to make real decisions.
Learning objectives
- Solve simultaneous linear equations by graphing, substitution and elimination.
- Explain why a solution is a point of intersection.
- Model choices such as competing price plans with a system.
AERO Mathematics alignment
AERO.M8.AE.4
Analyse and solve pairs of simultaneous linear equations graphically and algebraically, understanding that solutions correspond to points of intersection.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
Plan A: 40 QAR per month with no joining fee. Plan B: 10 QAR joining fee plus 30 QAR per month.
Think about it
After how many months do the two plans cost the same, and which is cheaper before that?
Hint: Write an equation for each and find where they meet.
EXPLAIN
Two equations, one point
A system of two linear equations asks for values that satisfy both at once. Graphically, the solution is the point of intersection.
| Method | How it works | Best when |
|---|---|---|
| Graphing | Draw both lines, read the intersection | You want a visual answer |
| Substitution | Replace one variable using the other equation | One equation gives y directly |
| Elimination | Add or subtract to remove a variable | Coefficients line up neatly |
Worked example
c = 40m and c = 30m + 10. Setting them equal gives 40m = 30m + 10, so 10m = 10 and m = 1 month, at a cost of 40 QAR. Before one month Plan B is cheaper; after it, Plan A is.
Three possible outcomes
- Lines cross once: exactly one solution.
- Lines are parallel: no solution.
- Lines coincide: infinitely many solutions.
Remember
Always substitute your answer back into both original equations. A point that satisfies only one of them is not a solution to the system.
INVESTIGATION
Try it yourself
Find two real mobile or streaming plans with different fixed fees and rates. Write the system, solve it algebraically, then graph both lines and mark the break-even point. Recommend a plan based on your own usage.
Watch
Solving Linear Systems by Substitution
Khan Academy · 9:00
A dependable algebraic method to back up your graphs.
Before you watch: What does the intersection point of two lines represent?
- Solve y = 2x + 1 and y = −x + 7.
- What does it mean when a system has no solution?
Interactive simulation · PhET
Graphing Lines
Graph two lines at once and read their intersection, then verify it algebraically.
While you explore
- What does the intersection point mean in a price-comparison problem?
- When do two lines have no intersection, and what does that mean for the system?
Key vocabulary
- Elimination
- Adding or subtracting equations to remove one variable.
- System of equations
- Two or more equations that must be true at the same time.
Practice questions
0/1 correct
Level 1 · Criterion A
Solve the system y = 2x + 1 and y = −x + 7.
MYP criterion tasks
Level 2 · Criterion D
Plan A costs 40 per month. Plan B costs 10 plus 30 per month. Find the break-even month and advise a user who stays 6 months.
Reflect & track
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