Systems of Equations — Substitution Method
SAT Relevance: Systems of equations appear on nearly every SAT Math section. You can expect 2–4 questions that test your ability to solve systems, and the substitution method is one of the fastest approaches when one variable is already isolated or easy to isolate.
What Is a System of Equations?
A system of equations is a set of two or more equations with the same variables. The solution is the ordered pair $(x, y)$ that makes both equations true simultaneously.
Geometrically, the solution is the point where the two lines intersect.
Example System:
$$ \begin{cases} y = 2x + 1 \\ 3x + y = 11 \end{cases} $$Our goal is to find the ordered pair \((x, y)\) that satisfies both equations simultaneously.
The Big Idea of Substitution
The substitution method works by:
- Isolate one variable in one equation.
- Substitute that expression into the other equation.
- Solve the resulting single-variable equation.
- Back-substitute to find the other variable.
💡 SAT Pro Tip: Use substitution when one equation already has a variable isolated (e.g., $y = \ldots$ or $x = \ldots$). This saves precious time!
Visualizing the Solution
The graph below shows two lines. Their intersection point is the solution to the system. Drag the intersection point to see the coordinates!
Interactive visualization — available in the full lesson