Step 1: Understand what the answer means
The answer is an ordered pair. It gives one value for x and one value for y.
x + y = 7
x - y = 1
solution = (x, y)
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Algebra
Geometry
Solve systems
Two equations, one shared answer
x + y = 7, x - y = 1
A simultaneous equation asks for values of x and y that make both equations true at the same time.
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Worked Examples
A simultaneous-equation solution must make both equations true at the same time. Different systems are easier with different methods.
Step 2: Use substitution when a variable is already isolated
If one equation already says what y equals, put that expression into the other equation.
y = 2x + 1
x + y = 10
x + (2x + 1) = 10
3x + 1 = 10
3x = 9
x = 3
y = 2(3) + 1 = 7
solution = (3, 7)
Step 3: Use elimination when variables line up
If the same variable appears in both equations, subtract or add the equations to remove it.
2x + y = 11
x + y = 7
(2x + y) - (x + y) = 11 - 7
x = 4
4 + y = 7
y = 3
solution = (4, 3)
Step 4: Use graphing to see the intersection
Each equation is a line. The crossing point gives the same x and y for both lines.
y = x + 1
y = -x + 5
x + 1 = -x + 5
2x = 4
x = 2
y = 3
intersection = (2, 3)
Step 5: Check the answer in both equations
Substitute the pair back into every equation. Both equations must be true.
2(4) + 3 = 11
4 + 3 = 7
(4, 3) works for both equations