What is a System of Equations?
A system of linear equations is a set of two or more equations containing the same variables (usually x and y).
Graphically, each equation forms a straight line on a coordinate grid. The solution to the system is the single point (x, y) where the two lines cross (intersect)!
When to Use Substitution vs. Elimination
| Condition | Recommended Method | Why? |
|---|---|---|
| One variable is already isolated (e.g. y = 3x - 4) | Substitution | Plug directly in with zero rearrangement |
| One variable has a coefficient of 1 or -1 (e.g. x + 4y = 9) | Substitution | Easy to isolate x = 9 - 4y without creating fractions |
| Both equations are in standard form (Ax + By = C) | Elimination | Coefficients can easily cancel out by adding or subtracting |
| Matching or opposite coefficients (e.g. +2y and -2y) | Elimination | Adding the equations eliminates the variable immediately |
Method 1: The Substitution Method
The Step-by-Step Game Plan:
- Isolate one variable in whichever equation is simplest.
- Substitute that expression into the other equation (use parentheses!).
- Solve the resulting single-variable equation.
- Back-substitute that number to find the second variable.
Worked Example 1: Solving with Substitution
Problem:
Solve the system:
Equation 1: y = 3x - 5
Equation 2: 2x + y = 10
Step 1: Substitute Equation 1 into Equation 2
Since Equation 1 already states y = (3x - 5), replace y in Equation 2:
Step 2: Combine like terms and solve for x
Add 5 to both sides:
Divide by 5:
Step 3: Find y using back-substitution
Plug x = 3 into Equation 1:
Step 4: Write as an ordered pair and check
Solution: (3, 4)
Verification Check:
- •Eq 1: 4 = 3(3) - 5 ⟹ 4 = 4 (True)
- •Eq 2: 2(3) + 4 = 10 ⟹ 6 + 4 = 10 (True)
Method 2: The Elimination Method (Addition Method)
In the elimination method, you add or subtract the two equations together so that one variable adds to zero and disappears!
Worked Example 2: Solving with Elimination
Problem:
Solve the system:
Equation 1: 3x + 2y = 16
Equation 2: 5x - 2y = 0
Step 1: Notice matching opposite coefficients
The y terms are +2y and -2y. Adding them together will cancel out y completely!
Step 2: Add the two equations vertically
Step 3: Back-substitute to find y
Substitute x = 2 into Equation 2:
Final Solution: The lines intersect at (2, 5).
What If Coefficients Don't Match? (Multiplying First)
If neither variable matches, multiply one or both equations by a constant number first:
Multiply the second equation by 3:
Now add to the first equation:
Special Cases: No Solution vs. Infinitely Many Solutions
Not all systems have exactly one solution:
- •No Solution (Parallel Lines): The lines never cross. When solving, variables cancel out and leave an impossible statement like 0 = 7.
- •Infinitely Many Solutions (Same Line): Both equations represent the exact same line. Variables cancel out leaving a universal truth like 0 = 0.
Always Distribute Minus Signs Carefully!
When substituting an expression like y = (2x - 3) into 5x - y = 8, remember the minus sign applies to both terms inside:
5x - (2x - 3) = 5x - 2x + 3 = 8. Dropping the negative on the second term is the most common test mistake!