Simultaneous equations
Find a pair of values that satisfies two constraints at the same time.
Before you begin
- Linear equations
- Multiplying expressions
Interpret the common solution
A solution to two equations must satisfy both of them. Geometrically it is an intersection of their graphs. Two distinct parallel lines have no common solution; two equations representing the same line have infinitely many. In a context, define the variables and make sure the final values respect any restrictions, such as a nonnegative item count.
Eliminate one variable
Make the coefficient of one variable equal in both equations, then subtract to eliminate it. Alternatively, make opposite coefficients and add. Multiply every term when scaling an equation. After finding the remaining variable, substitute it into either original equation and then check both originals.
Worked example
Solve x + y = 9 and 2x + y = 13.
Show the worked solution
- Subtract the first equation from the second: x = 4.
- Substitute into x + y = 9 to get y = 5.
- Check 4 + 5 = 9 and 2 × 4 + 5 = 13.
x = 4, y = 5.
Try it yourself
For x + y = 10 and x − y = 2, what is x?
Common mistakes
- Subtract the whole equation, including a negative term.
What can you explain now?
Solve 3x + y = 17 and x + y = 7.
Compare with the explanation
x = 5, y = 2.
Subtract to obtain 2x = 10. Substitute x = 5 into x + y = 7. Both original equations then hold.
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Common Core mathematics standards ↗