Systems of Linear Equations
Systems of Linear Equations
A system of two linear equations in two variables:
Solving Methods
Substitution
- Isolate one variable in one equation
- Substitute into the other equation
- Solve → back-substitute
Best when one variable already has coefficient 1 or is already isolated.
Example:
Elimination (Addition)
- Multiply one or both equations so a variable has matching (or opposite) coefficients
- Add or subtract equations to eliminate that variable
- Solve → back-substitute
Best when both equations are in standard form.
Example:
Number of Solutions
| Condition | Solutions | Geometrically |
|---|---|---|
| One (unique) | Lines intersect | |
| None | Lines are parallel | |
| Infinite | Same line |
Quick Check (Slope Comparison)
Convert both to slope-intercept form:
- Different slopes → one solution
- Same slope, different intercept → no solution
- Same slope, same intercept → infinite solutions
Systems with a Parameter
"For what value of
No solution when slopes are equal but intercepts differ:
Word Problems Strategy
- Define variables — write what
- Write two equations from two different conditions
- Solve using substitution or elimination
- Check the answer makes sense in context
Common Setups
| Scenario | Equations |
|---|---|
| Mixture problems | |
| Speed/distance | |
| Cost problems | quantity equation + value equation |
Example:
"Tickets cost $8 for adults and $5 for children. 200 tickets sold totaling $1,300."
SAT Tips
- Speed matters — if one equation already has a variable isolated, use substitution; otherwise prefer elimination
- "How many solutions?" — don't solve, just compare slopes (coefficients)
- Systems with fractions — multiply to clear denominators first
- The SAT sometimes gives systems disguised as word problems — look for two unknowns and two conditions
- Check your answer — plug the solution back into BOTH original equations
