Systems of 2 Linear Equations in 2 Variables
Systems of equations show up on nearly every test, often disguised as word problems about tickets, money, or mixtures. Knowing two clean methods lets you crush these in under a minute.
Watch & learn
Short walkthroughs on Systems of 2 Linear Equations in 2 Variables — ours, plus curated picks. Watch one if a written explanation isn't clicking, or after to reinforce.
A system of two linear equations is just two equations that share the same two variables (usually x and y). A solution is the pair of values (x, y) that makes both equations true at the same time. Graphically, each linear equation is a straight line, and the solution is the point where the two lines cross.
Description of the figure:
- The x-axis runs from negative 2 to 8 and the y-axis from negative 2 to 12, with grid lines every 1 unit.
- The first line rises from left to right.
- It passes through the origin at (0, 0).
- It also passes through (negative 2, negative 2) and (8, 8).
- The second line falls from left to right.
- It crosses the y-axis at (0, 12) and meets the x-axis at (6, 0).
- It also passes through (1, 10).
- The two lines intersect at (4, 4).
- Marked point: (4, 4).
Each equation is a line — here y = x and y = −2x + 12. The solution, (4, 4), is where they intersect.
Description of the figure:
- The x-axis runs from 0 to 14 and the y-axis from negative 2 to 14, with grid lines every 1 unit.
- The first line falls from left to right.
- It crosses the y-axis at (0, 14) and meets the x-axis at (14, 0).
- It also passes through (1, 13).
- The second line rises from left to right.
- It meets the x-axis at (6, 0).
- It also passes through (4, negative 2) and (14, 8).
- The two lines intersect at (10, 4).
- Marked point: (10, 4).
Solving x + y = 14 and x − y = 6: the two lines intersect at (10, 4).
How to classify the number of solutions.
Key reflex: when you see two equations with two variables, ask "Can I add/subtract to kill a variable? If not, can I multiply first, or substitute?"
Check your understanding with a question from this topic:
If the system x + y = 14 and x - y = 6 is solved, what is the value of x?
Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).
Worked examples
If 2x + y = 11 and x - y = 1, what is the value of x?
A theater sells adult tickets for $12 each and child tickets for $8 each. On Saturday, the theater sold 150 tickets and collected $1,480 in revenue. How many adult tickets were sold?
In the system below, c is a constant. If the system has no solution, what is the value of c?
cx + 4y = 9 3x + 2y = 5
Common pitfalls
Questions often ask for x, y, or even x + y — and the trap answers include the other value. After solving, re-read what's actually requested before bubbling.
When you scale an equation to set up elimination, every term — including the constant on the right — must be multiplied. Multiplying only the variable terms gives a wrong system.
Both have equal variable coefficients (parallel-looking). If the constants also match, it's the same line (infinite solutions); if the constants differ, the lines are parallel (no solution).
Subtracting flips every sign in the second equation. (2x+3y) - (2x+2y) is +y, not +5y. When in doubt, multiply by -1 and add instead.
Key takeaways
A solution to a system is the (x, y) point where both equations are true — graphically, where the lines intersect.
Elimination: add or subtract (after scaling if needed) to cancel a variable; substitution: isolate one variable and plug it in.
Word problems = translate two facts into two equations, then solve.
One solution = different slopes; no solution = parallel (same slope, different intercept); infinitely many = identical equations.
Always answer the exact quantity asked — it may be x, y, or a combination.
Skill check
10 questions on Systems of 2 Linear Equations in 2 Variables, from our question bank. Score 70% or better across 10 answers to reach Silver and pass this step.