System of Equations Basics

Many word problems describe two unknown quantities linked by two separate conditions. Writing one equation per condition produces a system of equations, and two standard methods — substitution and elimination — turn that system into a single equation in one variable, which can then be solved directly.

What you will learn

• Set up a system of equations from a word problem
• Solve a system by substitution
• Solve a system by elimination

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

Turning a word problem into a system

A system of equations problem always has two unknown quantities and two independent pieces of information relating them. The first step is to name the unknowns with variables — for instance, letting $x$ and $y$ stand for the two numbers, ages, or prices described. The second step is to translate each condition, one sentence at a time, into an equation using those variables.

"The sum of two numbers is 15" becomes $x+y=15$. "One number is twice the other" becomes $x=2y$. Each condition alone leaves both unknowns undetermined, but together the two equations pin down exactly one pair of values (in most well-posed problems). Writing both equations clearly before attempting to solve either one prevents combining information incorrectly.

Solving by substitution

Substitution works by isolating one variable in one equation, then replacing that variable everywhere it appears in the other equation. If $x=2y$, then wherever $x$ appears in the equation $x+y=15$, it can be replaced by $2y$, giving $2y+y=15$, an equation in $y$ alone. Solving gives $y=5$, and substituting back into $x=2y$ gives $x=10$.

Substitution is most efficient when one equation already isolates a variable, or can be isolated with little work — for example, when one equation states that one quantity equals an expression in the other, as in age or ticket-count problems where one quantity is described directly in terms of another.

Solving by elimination

Elimination works by adding or subtracting the two equations (or multiples of them) so that one variable cancels out entirely. If $x+y=22$ and $x-y=6$, adding the two equations directly cancels $y$: $(x+y)+(x-y)=22+6$, giving $2x=28$, so $x=14$, and then $y=22-14=8$.

When the variable to eliminate does not already have matching (or opposite) coefficients, multiply one or both equations by a constant first so that it does. Elimination is usually faster than substitution when both equations are already written in the same form, such as $ax+by=c$, since no variable needs to be isolated before combining the equations.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Writing only one equation from a word problem when two separate conditions were actually given.
• Substituting an expression into the same equation it came from instead of into the other equation.
• Adding two equations to eliminate a variable when the coefficients only match after a required multiplication step.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• Translate each separate condition in a word problem into its own equation to build a system of two equations in two unknowns.
• Substitution isolates one variable and replaces it in the other equation; it works best when one equation already isolates a variable.
• Elimination adds or subtracts (possibly scaled) equations to cancel a variable directly; it works best when both equations share the same form.