Systems of Equations Worksheets

What is a system of equations?

A system of equations is two (or more) equations that share the same variables. On these worksheets each problem is a pair of linear equations in $x$ and $y$, such as $x + 2y = 8$ and $x - y = 2$. A solution is an ordered pair $(x, y)$ that makes both equations true at the same time. On a graph, it is the point where the two lines cross.

The worksheets practise the elimination method: add or subtract the equations so that one variable disappears, solve for the other variable, then substitute back. Every answer on the worksheets is a pair of whole numbers, so students can focus on the method rather than on fraction arithmetic.

• Easy — both equations start with $x$, so subtracting one equation from the other removes $x$ right away.
• Medium — all coefficients are positive, but usually one equation has to be multiplied before the subtraction works.
• Hard — coefficients can be negative, and usually both equations need to be multiplied.

Worked examples

Easy: $x + 2y = 8$ and $x - y = 2$

Both equations have exactly one $x$, so subtract the second equation from the first:

Substitute $y = 2$ into $x - y = 2$: $x - 2 = 2$, so $x = 4$. Answer: $x = 4,\ y = 2$.

Medium: $3x + 2y = 16$ and $x + 4y = 12$

Neither variable cancels yet. Multiply the first equation by 2 so both equations contain $4y$:

Now subtract $x + 4y = 12$: $5x = 20$, so $x = 4$. Substitute into $x + 4y = 12$: $4 + 4y = 12$, so $y = 2$. Answer: $x = 4,\ y = 2$.

Hard: $4x - 3y = 6$ and $3x + 2y = 13$

The $y$-terms have opposite signs, so aim for $-6y$ and $+6y$ and then add. Multiply the first equation by 2 and the second by 3:

Adding gives $17x = 51$, so $x = 3$. Then $3(3) + 2y = 13$ gives $2y = 4$, so $y = 2$. Answer: $x = 3,\ y = 2$.

Check your answers

Substitute the pair into both original equations. For the hard example: $4(3) - 3(2) = 12 - 6 = 6$ and $3(3) + 2(2) = 9 + 4 = 13$. Both are true, so $(3, 2)$ is the solution.

Common mistakes

• Subtracting a negative term. In the easy example, $2y - (-y) = 3y$, not $y$. Put the second equation in parentheses before subtracting.
• Multiplying only part of an equation. When you multiply an equation, every term changes, including the number on the right: $2(3x + 2y) = 2(16)$ gives $6x + 4y = 32$.
• Stopping after one variable. Finding $x = 3$ is only half the answer; the solution is the pair $(x, y)$.
• Checking only one equation. A wrong pair can still satisfy one of the equations. Always check both.

Sample problems

These problems come from the same generator as the worksheet above, three at each difficulty level. Press Generate for a fresh set.

Easy

1. $x + y = 13$
   $x - 2y = 1$
2. $x + 3y = 23$
   $x - 2y = -7$
3. $x + 3y = 27$
   $x - y = 3$

Medium

4. $5x + y = 27$
   $x + 3y = 11$
5. $2x + y = 5$
   $x + 2y = 4$
6. $4x + 2y = 16$
   $x + 4y = 18$

Hard

7. $2x + 4y = 18$
   $5x + 2y = 29$
8. $4x + 5y = 30$
   $3x + 4y = 23$
9. $3x - 5y = -27$
   $5x - 3y = -13$

Tips for teachers and parents

• Start with Easy so students see elimination work in one step before they have to plan which equation to multiply.
• On Medium and Hard, ask students to write down which variable they will eliminate and what they will multiply by before doing any arithmetic.
• Graphing one or two problems on grid paper shows why the answer is a single point where the lines meet.
• Substitution also works on every problem. Solving the same system both ways is a good check and a good discussion.

Related topics

• Solving Equations Algebra
• Quadratic Equations Algebra 2
• Functions Pre-Calculus