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How to Solve Systems of Equations
Three methods for finding values that make several equations true at the same time.
A system of equations is a set of equations that share the same unknowns. A solution is a set of values that satisfies every equation at once. For two equations in x and y, each equation is a line or curve, and the solutions are the points where the graphs meet.
Method 1: Substitution
Solve one equation for one variable, then substitute that expression into the other equation. Use substitution when one equation already has a variable on its own.
Example: Solve y = 2x + 1 and 3x + 2y = 16.
Replace y in the second equation: 3x + 2(2x + 1) = 16, so 7x + 2 = 16 and x = 2.
Put x = 2 back into y = 2x + 1 to get y = 5. The solution is (2, 5).
Try it in CasCalc →Method 2: Elimination
Add or subtract the equations so that one variable cancels. You may need to multiply one or both equations by a number first so that the coefficients match.
Example: Solve x + y = 10 and x − y = 4.
Add the equations: the y terms cancel, leaving 2x = 14, so x = 7. Then 7 + y = 10 gives y = 3. The solution is (7, 3).
Try it in CasCalc →Example with multiplying: Solve 2x + 3y = 12 and 4x − y = 10.
Multiply the second equation by 3: 12x − 3y = 30. Add it to the first: 14x = 42, so x = 3.
Then 4(3) − y = 10 gives y = 2. The solution is (3, 2). Check: 2(3) + 3(2) = 12 and 4(3) − 2 = 10.
Try it in CasCalc →Method 3: Graphing
Graph both equations and find the point where they cross. Graphing is less precise by hand, but it shows at a glance how many solutions there are. In CasCalc, put each equation on its own line and the graphs appear together, with intersection points marked.
Graph y = 2x + 1 →No solution or infinitely many
Not every system has exactly one answer:
| What happens | Graph | Number of solutions |
|---|---|---|
| You get one value for each variable | Lines cross once | One |
| You get a false statement, like 0 = 4 | Parallel lines | None |
| You get a true statement, like 0 = 0 | Same line | Infinitely many |
For example, y = 2x + 1 and y = 2x − 3 have the same slope but different intercepts. Subtracting them gives 0 = 4, which is impossible, so there is no solution.
A nonlinear system
Substitution also works when one equation is a curve.
Example: Solve x² + y² = 25 and y = x + 1.
Substitute: x² + (x + 1)² = 25, so 2x² + 2x − 24 = 0, or x² + x − 12 = 0.
That factors as (x + 4)(x − 3) = 0, so x = 3 or x = −4. The line meets the circle at (3, 4) and (−4, −3).
Graph the circle →Which method should you choose?
- Substitution when a variable is already isolated, or for nonlinear systems.
- Elimination when both equations are in the form ax + by = c.
- Graphing to see how many solutions to expect and to check your answer.
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