8. Linear Equations & Inequalities
Middle SchoolA linear relationship changes at a constant rate — the simplest and most useful pattern in mathematics. Here we solve linear equations, handle inequalities and their sign rule, and solve two equations at once.
Solving by balance
A linear equation in one variable has the form ax + b = 0 with a ≠ 0, and exactly one solution x = −b/a. The golden rule: whatever you do to one side you must do to the other, keeping the equation balanced. Permitted moves are adding/subtracting the same quantity and multiplying/dividing both sides by the same non-zero quantity.
An inequality (<, ≤, >, ≥) is solved the same way, with one crucial extra rule:
A system of two linear equations describes two lines; its solution is their intersection point. Three outcomes are possible: one solution (lines cross), no solution (parallel), or infinitely many (same line).
The scales in your mind
Picture an equation as a balance scale: the = sign is the pivot. Anything you add to the left pan you must add to the right, or the scale tips and the statement stops being true. Solving is simply peeling away everything around x, in reverse order of operations, until x stands alone.
Why does the inequality sign flip on a negative? Because negating reflects numbers across zero, reversing their order: 3 < 5, but −3 > −5. Multiplying by a negative is exactly such a reflection, so the "smaller/larger" relationship swaps. For systems, two intuitions help: substitution (solve one equation for a variable, plug it into the other) and elimination (add or subtract the equations to cancel a variable). Both reduce two unknowns to one.
Worked example: solve a system by elimination
Solve 3x + 2y = 16 and 5x − 2y = 8.
- The y-terms are +2y and −2y — opposite. Add the two equations to eliminate y: (3x + 5x) + (2y − 2y) = 16 + 8.
- This gives 8x = 24, so x = 3.
- Substitute x = 3 into the first equation: 3(3) + 2y = 16 → 9 + 2y = 16 → 2y = 7 → y = 3.5.
- Check in the second: 5(3) − 2(3.5) = 15 − 7 = 8. ✓
Answer: x = 3, y = 3.5 — the point where the two lines cross.
Test your understanding
Solve 2(x − 3) = 4x + 2 for x.
Expand: 2x − 6 = 4x + 2. Gather x-terms on one side: subtract 2x → −6 = 2x + 2. Subtract 2 → −8 = 2x, so x = −4. Always verify: 2(−4 − 3) = −14 and 4(−4) + 2 = −14. ✓
When does a system of two linear equations have no solution?
When the lines are parallel but distinct — same slope, different intercept, so they never meet. Algebraically, elimination collapses to a false statement like 0 = 5. If instead it collapses to a true statement like 0 = 0, the lines are identical and there are infinitely many solutions.
Solve the inequality −3x + 1 ≥ 7.
Subtract 1: −3x ≥ 6. Divide by −3 and flip the sign: x ≤ −2. Sanity check with x = −5: −3(−5) + 1 = 16 ≥ 7. ✓ And x = 0 gives 1, which is not ≥ 7 — correctly excluded.
A taxi charges $3 to start plus $2 per km. Write and use a linear model for a 12 km trip.
Cost C = 3 + 2d, where d is distance in km. The 3 is the intercept (fixed cost) and 2 is the slope (rate). For d = 12: C = 3 + 2(12) = $27. To find how far $50 buys, solve 50 = 3 + 2d → d = 23.5 km.
How the ideas connect
Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.
The process, step by step
Worked problems, step by step
Follow each solution line by line, then try to reproduce it on paper before moving on.
Example 1Solve 3x + 4 = 19.
- Subtract 4 from both sides: 3x = 15
- Divide both sides by 3: x = 5
- Check: 3(5)+4 = 19
Example 2Solve the system x + y = 7 and x - y = 1.
- Add the equations: 2x = 8 so x = 4
- Substitute into x + y = 7: y = 3
- Solution is x = 4, y = 3
Now you try
Work each one out first, then tap to reveal the worked answer.