Systems of Equations
Review solving systems by graphing, substitution, and elimination, plus how to classify solutions.
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Questions Covered in This Set
11 cards to master
What is a system of equations?
Two or more equations that must be true at the same time; the solution is the ordered pair (x, y) satisfying all of them.
Geometrically, what is the solution to a system of two linear equations?
The point where the two lines intersect.
When is substitution the best method?
When one variable is already isolated or easy to isolate (e.g., y = 3x − 5).
When is elimination the best method?
When both equations are in standard form Ax + By = C, so adding or subtracting cancels a variable.
What is the main drawback of solving by graphing?
It's unreliable when the intersection isn't at a neat grid point, like (2.37, 5.81); it's best for intuition and checking.
Solve: y = 15x + 40 and y = 25x
15x + 40 = 25x → 10x = 40 → x = 4, y = 100. Solution (4, 100): both gyms cost $100 after 4 months.
What does it mean if solving a system gives a false statement like 0 = 7?
No solution — the lines are parallel and the system is inconsistent.
What does it mean if solving a system gives 0 = 0?
Infinitely many solutions — both equations describe the same line; the system is dependent.
Solve by elimination: 3x + 2y = 16 and 5x − 2y = 8
Add: 8x = 24 → x = 3; then 3(3) + 2y = 16 → y = 3.5. Solution (3, 3.5).
Solve by substitution: x + 2y = 11 and 3x − y = 5
y = 3x − 5 → x + 2(3x − 5) = 11 → 7x = 21 → x = 3, y = 4. Solution (3, 4).
How do you check a solution to a system?
Substitute the ordered pair back into the other (both) original equation(s) and verify both are true.
