Algebra · Grade 8
Systems of Linear Equations
A system of linear equations is a set of equations that share the same variables. A solution makes every equation true at the same time.
Learning time: 20–25 minutes · Level: Developing
Learning goals
- Understand the key terms and concepts.
- Apply a reliable method to worked and new problems.
- Check your reasoning and communicate your answer.
1. What a solution means
The solution to two linear equations is an ordered pair (x, y) that satisfies both equations. On a graph, it is the intersection point of the two lines.
2. Solve by graphing
Graph both equations on the same coordinate plane. The point where the lines cross is the solution. Parallel distinct lines have no solution; the same line has infinitely many.
3. Substitution
Rearrange one equation to make a variable the subject, then substitute that expression into the other equation. Solve the resulting one-variable equation and substitute back.
4. Elimination
Write equations in aligned form. Add or subtract them to eliminate one variable. If coefficients do not match, multiply one or both equations first.
5. Check and interpret
Substitute the ordered pair into both original equations. In word problems, define variables, form two equations, solve, and explain what the values mean.
Choose substitution or elimination
A solution to a system is an ordered pair that makes both equations true. Substitution is useful when one equation already has a variable isolated. Elimination is useful when adding or subtracting equations can remove one variable. A system may have one solution, no solution (parallel lines), or infinitely many solutions (the same line).
Worked reasoning: Solve x + y = 9 and x − y = 3. Add the equations: 2x = 12, so x = 6. Substitute into x + y = 9: 6 + y = 9, so y = 3. Check the second equation: 6 − 3 = 3. The solution is (6, 3).
On a graph, the solution is the point where the two lines intersect. If the lines never meet, there is no solution; if they lie on top of one another, every point on the line is a solution.
- Keep equations aligned when adding or subtracting.
- Substitute the ordered pair into both original equations.
- State the solution as (x, y), not as two unrelated values.
Worked examples
- y = x + 2 and y = 5: x + 2 = 5, so x = 3 and y = 5.
- x + y = 10 and x − y = 2: add to get 2x = 12, so x = 6 and y = 4.
- y = 2x and x + y = 9: x + 2x = 9, so (x, y) = (3, 6).
Independent practice
Attempt each problem before checking the worked answer.
- Solve x + y = 8 and x − y = 2. Answer: (5, 3).
- Solve y = x + 1 and y = 7. Answer: (6, 7).
- Solve 2x + y = 11 and x + y = 7. Answer: (4, 3).
- Solve x + y = 12 and x = 5. Answer: (5, 7).
- How many solutions do distinct parallel lines have? Answer: None.
- How many solutions do two identical lines have? Answer: Infinitely many.
- Solve 3x + y = 13 and x + y = 7. Answer: (3, 4).
- Two adult tickets and one child ticket cost $25; one adult and one child cost $15. Find each price. Answer: adult $10, child $5.
Knowledge check
Choose one answer for each question, then check your answers for explanations.
Printable worksheet and answer key
Use the independent-practice list above as your worksheet. Show working, label units where appropriate, and review the answers after attempting all questions.
Study reminder
Write the rule or method used beside each solution. This helps identify and correct mistakes.