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A system of two linear equations is just two lines. The solution is the single point that makes both equations true, which on a graph is exactly where the two lines cross. Hold onto that picture and the algebra has a purpose.
Key takeaways
- The solution is the (x, y) that satisfies both equations, the intersection point.
- Solve by substitution or elimination; both find that point.
- No solution = parallel lines (same slope, different intercept).
- Infinite solutions = the same line written two ways.
The picture
Take y = 2x + 1 and y = -x + 7. Two lines, one crossing point, and that point is the answer to the system. (On the real test, the built-in Desmos calculator will show you this picture in seconds.)
Try one
If y = 2x + 1 and y = -x + 7, what is the solution (x, y) to the system?
The two methods
- 1
Substitution
When one equation is already solved for a variable, plug it into the other. (Best when you see y = \dots.)
- 2
Elimination
Add or subtract the equations to cancel a variable. (Best when terms line up, like 3x + 2y and 3x - 2y.)
- 3
Back-substitute
Found one variable? Plug it back to get the other, then check both equations.
Same slope, different intercept → no solution (parallel). Same slope and intercept → infinitely many (one line). On the SAT, matching slopes is the tell.
The no-solution question, worked
The SAT’s favorite hard version of this skill hides the special case inside a constant: "for what value of a does the system have no solution?" That’s really asking what makes the lines parallel. Make the x and y coefficients proportional, and check the constants break the pattern.
In the system 3x - 6y = 5 and ax + 2y = 7, where a is a constant, for what value of a does the system have no solution?
Scaling an equation to line up coefficients for elimination? The constant comes too: doubling 2x + y = 5 gives 4x + 2y = 10, not 4x + 2y = 5. Scaling the left side and forgetting the right is the quiet slip behind most wrong elimination answers.
When you see it, do this
| The question shows… | The move |
|---|---|
| One equation already reads y = mx + b | Substitute it into the other equation |
| Coefficients that match or are opposites | Add or subtract the equations to eliminate |
| "What is x + y?" (a combination, not each variable) | Add the equations as-is and scale; don’t solve for x and y separately |
| "For what value … no solution?" | Set the coefficient ratios equal and the constants unequal |
Solve systems with a tutor that ties the algebra back to the graph.
Practice Systems of Equations →Frequently asked questions
- What is the solution to a system of equations?
- It’s the (x, y) pair that satisfies both equations at once: graphically, the point where the two lines intersect.
- When does a system have no solution or infinite solutions?
- No solution when the lines are parallel (same slope, different y-intercept). Infinitely many solutions when both equations describe the same line (same slope and intercept).
- Should I use substitution or elimination on the SAT?
- Use substitution when one equation is already solved for a variable; use elimination when terms line up so adding or subtracting cancels a variable. Both give the same answer. Pick the faster one for the problem.
- Do I always have to solve for both variables?
- No. When the question asks for a combination like x + y, try adding or subtracting the equations first. SAT systems are often built so the combination falls out in one step, with no full solve required.