How can I solve a two-equation system by substitution?
Quick answer
Solve one equation for a variable, replace that variable in the other equation with its equivalent expression, and solve the resulting one-variable equation. Substitute that value back to find the second variable. Write the solution as an ordered pair, then check it in both original equations. Choose an equation and variable that make the rearranging and arithmetic manageable. [S1]
The question, the details, the next step.
What does solving by substitution mean, and how do I start?
Substitution means replacing a variable in one equation with an expression that has the same value, taken from the other equation. That turns the system into an equation with just one variable. After solving it, you use the value you found to get the other variable. [S1]
Start by looking for an equation where one variable is already alone, or can be isolated with little work. For example, in the system
y = 2x + 1
x + y = 10
y is already expressed in terms of x. So, in the second equation, replace y with 2x + 1. The replacement is valid because the first equation says those expressions have the same value. Keep both original equations handy: you will need them for the final check.
Can you work through that example without skipping the steps?
Begin with the system:
y = 2x + 1
x + y = 10
- Replace y in the second equation with
2x + 1:x + (2x + 1) = 10. - Combine like terms:
3x + 1 = 10. - Subtract 1 from both sides:
3x = 9. - Divide both sides by 3:
x = 3. - Substitute
x = 3into the first equation:y = 2(3) + 1 = 7.
The ordered pair is (3, 7), with x listed first and y second. Substitution reduces the system to one variable; substituting the result back gives the second value. [S1]
How do I know whether the answer is actually right?
Check the pair in each original equation, not just the equation you simplified. For (3, 7), the first equation gives 7 = 2(3) + 1, or 7 = 7. The second gives 3 + 7 = 10, also true. Since the pair satisfies both originals, it passes the check. [S1]
This check can catch an arithmetic or sign error made during substitution. Keep the original equations visible until both checks are complete.
What if neither variable is already alone, and what should I write down to practice?
Choose either equation and rearrange it to isolate one variable; then substitute that equivalent expression into the other equation. If both choices look similarly easy, pick the one with fewer steps or simpler arithmetic. The method is to isolate a variable, substitute, solve the one-variable equation, substitute back, and check the ordered pair in both originals. [S1]
A reusable practice layout is:
Original system: __________________ and __________________
Isolate: __________ = __________________
Substitute: ____________________________________________
Solve: __________ = ______
Substitute back: __________ = ______
Ordered pair: (______, ______)
Check in equation 1: _________________________________
Check in equation 2: _________________________________
For the worked example, the checks are 7 = 2(3) + 1 and 3 + 7 = 10. If a check fails, review the rearranging, substitution, and arithmetic. This procedure is for two-variable linear systems. [S1]
A practical checklist
Copy both original equations so you can check the final pair.
Choose an equation and variable that are easy to isolate.
Replace that variable in the other equation with its equivalent expression.
Solve the resulting one-variable equation, then substitute back.
Write the answer as an ordered pair and check it in both original equations.
A few more useful answers
Do I substitute into the equation I used to isolate the variable?
Usually, substitute into the other equation to create a one-variable equation. Once you have one value, use either original equation to find the remaining value.
Why should I check both original equations?
A pair is a solution to the system only if it makes both equations true. Checking both can catch mistakes made while rearranging, substituting, or calculating.
What does the ordered pair list first?
For variables x and y, write the x-value first and the y-value second: (x, y). In the example, that is (3, 7).
Sources
References used for this guide.
- [S1] OpenStax Elementary Algebra 2e: Solving Systems of Equations by Substitution openstax.org · Checked