the whole idea, in one sentence
When two equations are true at the same time, their solution is the point where their graphs cross — and there are three ways to find it: graph, substitute, or eliminate.
All three teach the same idea from a different angle.
Your two best friends, Mia and Jada, won't tell you their ages. But they give you two clues:
"Together we're 30."
"Mia is 4 years older than Jada."
Two unknowns: Mia's age (m) and Jada's age (j). Two clues = two equations.
m + j = 30
m = j + 4
This is a system of equations. The answer has to make BOTH true at the same time.
Try it: if Jada is 13, Mia is 17. Together = 30 ✓. Mia is 4 older than Jada ✓. So Jada = 13, Mia = 17.
The big idea: any time you have two clues about two unknowns, you can find both — as long as the clues actually contain enough information. The whole unit is just three methods for cracking these.
Each equation in a system is a line on a graph. Two equations = two lines.
The solution is wherever those lines cross. That single point (x, y) is the only place where both equations are true at once.
y
↑
| \ /
| \ /
| \ / ← they cross HERE = (3, 4)
| / \
| / \
|————————→ x
two lines that cross at a single point have one solution
Three possible outcomes when you graph a system:
Cross once → one solution (most common)
Parallel (never touch) → no solution
Same line (overlap completely) → infinite solutions
Graphing is great for seeing what's happening, but it can be hard to read exact answers off a graph — especially if the crossing isn't on a whole-number point. That's why we also have substitution and elimination: they give exact answers.
Three methods. Pick whichever feels easiest for the problem in front of you.
METHOD 1 — Graphing. Graph both lines on the same axes. Find where they cross. That (x, y) is the answer. Good for visual people; not great if the answer isn't a whole-number point.
METHOD 2 — Substitution. If one equation is solved for a variable (like y = 2x + 1), substitute that whole expression into the OTHER equation.
Example:
y = 2x + 1
3x + y = 11
Step 1. Replace y in the second equation with "2x + 1":
3x + (2x + 1) = 11
Step 2. Solve for x: 5x + 1 = 11 → 5x = 10 → x = 2.
Step 3. Plug x = 2 back into y = 2x + 1: y = 5.
Answer: (2, 5).
METHOD 3 — Elimination. Add or subtract the two equations so one variable cancels out.
Example:
x + y = 10
x − y = 4
Step 1. Add the two equations straight down. The y's cancel: (x + x) + (y − y) = 14 → 2x = 14.
Step 2. Solve: x = 7.
Step 3. Plug x = 7 into either equation. 7 + y = 10 → y = 3.
Answer: (7, 3).
Sometimes you have to multiply one equation by a number first to make a variable cancel. That's still elimination — just one extra prep step.
Rule of thumb: If one equation already has a variable solved (like y = …), use substitution. If both equations are in ax + by = c form, elimination is usually faster.
5 problems. Type your answer as an ordered pair like (2,5) — no spaces. Use whichever method feels easiest.