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Systems of Equations: Solving Two Equations at Once

Learn to solve systems of equations three ways: substitution, elimination, and graphing — with worked examples for each method.

Practice Systems of Equations

What Are Systems of Equations?

A system of equations is just two (or more) equations that share the same variables, where you're looking for the values that make all of them true at the same time. Graphically, that's the point where two lines cross. Algebraically, there are two main methods for finding it: substitution and elimination.

Method 1: Substitution

Solve one equation for one variable, then substitute that expression into the other equation.

Worked Examples

Substitution Method

Medium

y = x + 2 2x + y = 11

Step 1: Substitute the expression
2x + (x + 2) = 11
Since y = x + 2, we can replace y in the second equation with (x + 2).
Step 2: Simplify and solve for x
3x + 2 = 11
Combine like terms.
Step 3: Isolate x
3x = 9 -> x = 3
Subtract 2 from both sides, then divide by 3.
Step 4: Solve for y
y = 3 + 2 = 5
Substitute x = 3 back into the first equation to find y.
Final Answer:
(3, 5)

Method 2: Elimination

Add or subtract the equations to cancel out one variable.

Elimination Method

Hard

3x + 2y = 16 5x - 2y = 8

Step 1: Add the equations
(3x + 2y) + (5x - 2y) = 16 + 8
The +2y and -2y will cancel out (eliminate) each other when we add the equations together.
Step 2: Solve for x
8x = 24 -> x = 3
Simplify and divide by 8.
Step 3: Substitute x back in
3(3) + 2y = 16
Plug x = 3 into the first equation to find y.
Step 4: Solve for y
9 + 2y = 16 -> 2y = 7 -> y = 3.5
Subtract 9 and divide by 2.
Final Answer:
(3, 3.5)

Method 3: Graphing

Graph both equations as lines — the point where they intersect is the solution. This method is the most visual but the least precise unless the intersection lands on a clean grid point.

Special cases

  • No solution: the lines are parallel (same slope, different y-intercept) — they never cross.
  • Infinite solutions: both equations describe the same line — every point on it works.

Common mistakes to avoid

  • Forgetting to solve for the second variable. Finding x isn't the finished answer — plug it back in to get y too.
  • Sign errors during elimination. When subtracting equations, distribute the negative sign across the entire equation, not just the first term.
  • Mismatching coefficients before elimination. Sometimes you need to multiply one or both equations by a constant first so a variable's coefficients match (or are opposites) before adding or subtracting.

Where this shows up next

Systems of equations are the algebraic backbone of solving real-world problems with multiple unknowns — from mixture and rate problems in Algebra 2 to systems of linear equations in matrix form later in math.

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