Systems of Equations — Substitution and Elimination Methods

By MathPal Editorial Team·Published Apr 21, 2026

Grade: 8-9 | Topic: Algebra

What You Will Learn

A system of equations is two or more equations that must be true at the same time. In this guide you will learn to solve a system of two linear equations using three methods — graphing (conceptual overview), substitution, and elimination — and apply these methods to word problems.

Theory

What a solution means

The solution to a system of two equations is the point $(x, y)$ where both equations are satisfied simultaneously — the intersection of their graphs.

A system can have:

• One solution (the lines intersect at one point) — most common case.
• No solution (the lines are parallel — same slope, different intercepts).
• Infinitely many solutions (the lines are identical).

Method 1 — Substitution

Use substitution when one equation is easily solved for one variable.

Steps:

1. Solve one equation for $x$ or $y$.
2. Substitute that expression into the other equation.
3. Solve for the remaining variable.
4. Substitute back to find the first variable.
5. Check in both original equations.

Method 2 — Elimination (Addition/Subtraction)

Use elimination when the coefficients of one variable can be made equal.

Steps:

1. Multiply one or both equations so a variable has equal (and opposite) coefficients.
2. Add the equations to eliminate that variable.
3. Solve for the remaining variable.
4. Substitute back to find the other variable.
5. Check in both original equations.

Worked Examples

Example 1 — Substitution method

Solve the system:

$\begin{cases} y = 2x + 1 \\ 3x + y = 16 \end{cases}$

Step 1: The first equation already isolates $y$: $y = 2x + 1$.

Step 2: Substitute into the second equation:

$3x + (2x + 1) = 16$ $5x + 1 = 16$ $5x = 15$ $x = 3$

Step 3: Find $y$: $y = 2(3) + 1 = 7$.

Check: $3(3) + 7 = 9 + 7 = 16$ ✓.

Solution: $(3, 7)$

Example 2 — Elimination method

Solve the system:

$\begin{cases} 2x + 3y = 12 \\ 2x - y = 4 \end{cases}$

Step 1: The $x$-coefficients are both 2. Subtract the second equation from the first:

$(2x + 3y) - (2x - y) = 12 - 4$ $4y = 8$ $y = 2$

Step 2: Substitute into the second equation: $2x - 2 = 4 \Rightarrow 2x = 6 \Rightarrow x = 3$.

Check: $2(3) + 3(2) = 6 + 6 = 12$ ✓ and $2(3) - 2 = 4$ ✓.

Solution: $(3, 2)$

Example 3 — Elimination with multiplication

Solve the system:

$\begin{cases} 3x + 4y = 10 \\ x - 2y = -2 \end{cases}$

Step 1: Multiply the second equation by 2 to match the $y$-coefficient:

$2x - 4y = -4$

Step 2: Add to the first equation:

$(3x + 4y) + (2x - 4y) = 10 + (-4)$ $5x = 6$ $x = 1.2$

Step 3: Substitute: $1.2 - 2y = -2 \Rightarrow -2y = -3.2 \Rightarrow y = 1.6$.

Solution: $(1.2, 1.6)$

Example 4 — Word problem

Two numbers have a sum of 25 and a difference of 7. Find both numbers.

Step 1: Write the system. Let $x$ be the larger number and $y$ the smaller:

$\begin{cases} x + y = 25 \\ x - y = 7 \end{cases}$

Step 2: Add the equations: $2x = 32 \Rightarrow x = 16$.

Step 3: Substitute: $16 + y = 25 \Rightarrow y = 9$.

Answer: The numbers are 16 and 9.

Common Mistakes

Mistake 1 — Substituting into the same equation you solved

❌ After solving $y = 2x + 1$ from equation 1, substituting back into equation 1 to find $x$.

✅ Always substitute into the other equation. Substituting into the same equation gives an identity (e.g., $0 = 0$), not a solution.

Mistake 2 — Sign error when subtracting equations in elimination

❌ $(2x + 3y) - (2x - y)$: writing $3y - y = 2y$ but also writing $-y$ as $+y$ in the subtraction.

✅ Subtracting an equation means changing the sign of every term: $(2x - y) \to -2x + y$. So the combination gives $3y + y = 4y$.

Mistake 3 — Not checking the solution

❌ Stopping after finding $x$ and $y$ without verifying.

✅ Always substitute both values into both original equations. Both must be satisfied.

Practice Problems

Problem 1: Solve by substitution.

$\begin{cases} x = 3y - 1 \\ 2x + y = 12 \end{cases}$

Show Answer

Substitute: $2(3y-1) + y = 12 \Rightarrow 6y-2+y = 12 \Rightarrow 7y = 14 \Rightarrow y = 2$.

$x = 3(2)-1 = 5$.

Solution: $(5, 2)$

Problem 2: Solve by elimination.

$\begin{cases} x + y = 9 \\ x - y = 3 \end{cases}$

Show Answer

Add: $2x = 12 \Rightarrow x = 6$. Substitute: $6 + y = 9 \Rightarrow y = 3$.

Solution: $(6, 3)$

Problem 3: A cinema sold adult tickets for $12 and child tickets for $7. In one session 50 tickets were sold for a total of $460. How many adult and child tickets were sold?

Show Answer

Let $a$ = adults, $c$ = children.

$a + c = 50$ and $12a + 7c = 460$.

From the first: $a = 50 - c$. Substitute: $12(50-c) + 7c = 460 \Rightarrow 600 - 12c + 7c = 460 \Rightarrow -5c = -140 \Rightarrow c = 28$.

$a = 50 - 28 = 22$.

22 adult and 28 child tickets.

Problem 4: Solve by elimination.

$\begin{cases} 4x + 3y = 24 \\ 2x + y = 10 \end{cases}$

Show Answer

Multiply second by 3: $6x + 3y = 30$. Subtract first: $2x = 6 \Rightarrow x = 3$.

$2(3) + y = 10 \Rightarrow y = 4$.

Solution: $(3, 4)$

Summary

• A system of equations asks you to find values satisfying two equations simultaneously.
• Substitution: isolate one variable, substitute the expression into the other equation.
• Elimination: add or subtract equations (after scaling) to cancel one variable.
• A system has one solution (intersecting lines), no solution (parallel lines), or infinite solutions (same line).
• Always verify your solution by substituting into both original equations.

Related Topics

• Linear Equations — How to Solve Step by Step — foundational single-equation solving
• How to Graph Linear Equations — see where two lines intersect
• Writing and Solving Equations from Word Problems — setting up systems from real-world scenarios

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