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Stage 8 · Advanced Algebra Skills
Systems of Equations
A system is two equations with two variables — usually x and y. The solution is the one pair of values that makes both equations true at the same time. This lesson shows you how to find it by substitution and by elimination, and how to check a solution.
Solving a system is built entirely on skills you already have: solving 1- and 2-step equations and substituting values into expressions. If those feel shaky, spend a little time on them first — everything here will feel much easier.
A system of equations is two equations that share the same two variables, like x and y. On their own, each equation has infinitely many solutions. Together, they usually pin down exactly one pair — the point (x, y) where both are true. Finding that pair is the whole job.
{
x + y = 7
x − y = 3
The solution is (5, 2) — the only pair that works in both lines. We'll find it three different ways below.
Your answer comes in three flavors. The GED may ask for the full solution as a coordinate pair (x, y), just the value of x, or just the value of y. Solve the whole system, then report whichever piece the question asks for.
Method 1 — Checking a Solution
The simplest question type hands you a point and asks whether it solves the system. The rule is short: a real solution must make both equations true. Substitute the x and y into each equation; if even one comes out false, the pair is not the solution.
How to check a point (x, y)
1
Substitute the x-value and y-value into the first equation. Simplify. Is it true?
2
If yes, substitute the same pair into the second equation. Is it also true?
3
Both true → it's the solution. Either one false → it's not.
Is (5, 2) the solution?
x + y = 7x − y = 3
→ Step 1. Read the pair (5, 2) carefully. The first number is the x-value and the second number is the y-value, so x = 5 and y = 2.
→ Step 2. A real solution has to work in BOTH equations, so substitute x = 5 and y = 2 into each one — replace every x with 5 and every y with 2.
→ Step 3. Put x = 5 and y = 2 into the first equation, x + y = 7:
5 + 2 = 7 ✓
→ Step 4. Now put the same x = 5 and y = 2 into the second equation, x − y = 3:
5 − 2 = 3 ✓
Both equations are true → (5, 2) is the solution.
One equation is not enough. The point (6, 1) also satisfies x + y = 7, but 6 − 1 = 5, not 3 — so it fails the second equation. A pair that only works in one equation is a trap answer. Always test both.
Method 2 — Substitution
Substitution shines when one equation already has a variable by itself — like y = 2x or x = y + 2. You take that expression and drop it into the other equation, which leaves a single-variable equation you already know how to solve.
The substitution routine
1
Get a variable alone in one equation (often it already is).
2
Substitute that expression into the other equation. Now there's only one variable.
3
Solve that one-variable equation.
4
Back-substitute the value you found to get the other variable. Write it as (x, y).
Example — Substitution
y = x + 1x + y = 9
→ Step 1. Equation 1 already has y by itself: y = x + 1. That's the expression we'll substitute.
→ Step 2. In equation 2, replace y with (x + 1). Now there's only one variable left.
x + (x + 1) = 9
→ Step 3. Combine the like terms (x + x = 2x).
2x + 1 = 9
→ Step 4. Subtract 1 from both sides, then divide by 2 to solve for x.
2x = 8 → x = 4
→ Step 5. Back-substitute x = 4 into y = x + 1 to find y.
y = 4 + 1 = 5 → solution (4, 5)
Both equations solved for y? If you see y = 2x − 1 and y = x + 5, just set the two right sides equal: 2x − 1 = x + 5. They both equal y, so they equal each other.
Method 3 — Elimination
Elimination shines when the equations line up in x + y form. The idea: add or subtract the two equations so that one variable cancels out, leaving a single-variable equation. The cleanest case is when one variable already has opposite signs, like +y and −y — just add and it's gone.
The elimination routine
1
Line up the equations so x's are over x's and y's over y's.
2
Make one variable cancel. If the coefficients are opposites, add. If they match, subtract. If neither, multiply an equation by a number first.
3
Solve the single-variable equation that's left.
4
Back-substitute into either original equation for the other variable. Write it as (x, y).
Signs are opposite → ADD
+y and −y cancel when you add. x + y = 10 x − y = 4 add → 2x = 14.
Signs match → SUBTRACT
+y and +y cancel when you subtract. 5x + y = 17 2x + y = 8 subtract → 3x = 9.
Example — Elimination (multiply first)
3x + 2y = 16x + y = 6
→ Step 1. Look at the y-terms: 2y and y. They don't cancel yet, so make them match by multiplying the whole second equation by 2.
2(x + y) = 2(6) → 2x + 2y = 12
→ Step 2. Both equations now have 2y. Subtract the new equation from the first so the 2y terms cancel. The left side leaves x, and 16 − 12 leaves 4.
x = 4
→ Step 3. Back-substitute x = 4 into the simpler equation x + y = 6 to find y.
4 + y = 6 → y = 2 → solution (4, 2)
Which Method Should I Use?
Both methods always work — pick whichever fits the way the system is written. This table is your quick guide.
If the system looks like…
Reach for…
A variable is already alone (y = 2x, x = y + 2)
Substitution
Both equations are solved for y
Substitution (set them equal)
Lined up in x + y form with a variable that cancels
Elimination
You're handed a point to test
Checking (substitute into both)
Always finish by checking. Once you have an (x, y), plug it back into both original equations. If both are true, you're done and you can trust your answer — no second-guessing on test day.
⚠️ Common Mistakes
Almost every wrong answer on a system comes from one of these. The Guided Practice bank is built to train your eye for them.
Only checking one equation
A pair must satisfy both. Working in just one equation is the #1 trap answer.
Forgetting to find the second variable
You found x — but the question wanted (x, y). Back-substitute to get y before answering.
Swapping x and y in the pair
(5, 2) and (2, 5) are different points. Keep x first, y second.