Substitution Method

A system of equations is two equations with the same two variables, like $y = 3x + 1$ and $x + y = 9$ . Solving the system means finding the one pair of values — an $(x, y)$ point — that makes both equations true at the same time. On a graph, that pair is the point where the two lines cross.

Graphing works, but it's slow and only reliable when the answer lands on nice grid points. The two algebraic methods — substitution and elimination — get an exact answer every time. Each one turns the two-variable system into a single one-variable equation you already know how to solve.

What a solution to a system means

A solution must satisfy both equations, not just one. The pair $(2, 7)$ solves the system $y = 3x + 1$ and $x + y = 9$ because $7 = 3(2) + 1$ and $2 + 7 = 9$ are both true. Plenty of points satisfy one equation or the other; only the intersection point satisfies both.

That's also how you check your work: substitute your answer into both original equations. If either one fails, something went wrong along the way.

On a graph, each equation is a line, and the solution is the single point where they cross — the one place that lies on both lines at once. The two lines below meet at exactly one point.

Substitution: replace a variable

Substitution shines when one equation is already solved for a variable — something like $y = 3x + 1$ or $x = y - 4$ . Take that expression and substitute it in place of the variable in the other equation. Now the other equation has only one variable, and you solve it normally.

Once you have one variable's value, substitute it back into either original equation to find the other. Write the answer as an ordered pair $(x, y)$ — keep the $x$ -value first.

If neither equation is solved for a variable, look for one with a lone $x$ or $y$ (coefficient $1$ ) and solve for it first. From $x - y = 1$ , adding $y$ to both sides gives $x = y + 1$ , and now you can substitute.

Elimination: add the equations to cancel a variable

Elimination shines when both equations are lined up in the form $ax + by = c$. If one variable has opposite coefficients in the two equations — like $+3y$ and $-3y$ — add the equations together and that variable cancels, leaving one equation in one variable.

If nothing cancels yet, multiply one equation (or both) by a number chosen to create opposite coefficients. Multiply every term, including the right side — that's the step people rush and get wrong.

After solving for the surviving variable, substitute back into either original equation to find the other, and write the ordered pair.

Worked examples

Example 1: substitution, ready to go

Solve the system $y = 2x + 1$ and $x + y = 10$ .

The first equation gives

y

, so substitute

2x + 1

for

y

in the second

x + (2x + 1) = 10

Combine like terms

3x + 1 = 10

Subtract

1

from both sides

3x = 9

Divide both sides by

3

x = 3

Back-substitute to find

y

y = 2(3) + 1 = 7

Check in the second equation:

3 + 7 = 10

✓

Answer: $(3, 7)$

Example 2: substitution after solving for a variable

Solve the system $x - y = 2$ and $3x + y = 14$ .

Solve the first equation for

x

x = y + 2

Substitute into the second equation

3(y + 2) + y = 14

Distribute

3y + 6 + y = 14

Combine like terms

4y + 6 = 14

Subtract

6

, then divide by

4

y = 2

Back-substitute to find

x

x = 2 + 2 = 4

Answer: $(4, 2)$

Example 3: elimination, coefficients already opposite

Solve the system $2x + 3y = 12$ and $4x - 3y = 6$ .

The

y

-terms are

+3y

and

-3y

, so add the equations

6x = 18

Divide both sides by

6

x = 3

Substitute into the first equation

2(3) + 3y = 12

Subtract

6

, then divide by

3

y = 2

Check in the second equation:

4(3) - 3(2) = 12 - 6 = 6

✓

Answer: $(3, 2)$

Example 4: elimination with multiplication

Solve the system $3x + 2y = 16$ and $2x - y = 6$ .

Multiply the second equation by

2

so the

y

-terms become opposites

4x - 2y = 12

Add it to the first equation

7x = 28

Divide both sides by

7

x = 4

Substitute into

2x - y = 6

2(4) - y = 6

Solve for

y

y = 2

Check in the first equation:

3(4) + 2(2) = 12 + 4 = 16

✓

Answer: $(4, 2)$

Try one yourself

Common questions

How do I decide between substitution and elimination?

If either equation is already solved for a variable — or has a lone $x$ or $y$ with coefficient $1$ — substitution is usually fastest. If both equations are in $ax + by = c$ form, elimination is usually cleaner. Both methods always give the same answer, so pick whichever means less rearranging.

What does it mean if both variables cancel out?

You're left with a statement about numbers only. If it's false, like $0 = 5$ , the lines are parallel and the system has no solution. If it's true, like $0 = 0$ , the two equations describe the same line and there are infinitely many solutions.

Do I have to write the answer as an ordered pair?

Yes — a solution to a system is a point, so give both values as $(x, y)$ with the $x$ -value first. Writing only $x = 3$ is half an answer, and swapping the order changes which point you're naming.

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Substitution Method

What a solution to a system means

Substitution: replace a variable

Elimination: add the equations to cancel a variable

Worked examples

Example 1: substitution, ready to go

Example 2: substitution after solving for a variable

Example 3: elimination, coefficients already opposite

Example 4: elimination with multiplication

Try one yourself

Common questions

Keep learning

Slope-Intercept Form

The Slope Formula

Two-Step Equations