Systems of equations in two variables
Solving systems that pair a linear equation with a quadratic — the core of SAT Advanced Math systems questions.
Core Idea
Substitution collapses a two-variable system to one equation in one variable. The shape of that resulting equation — linear or quadratic — tells you how many solutions to expect.
Understanding
On the SAT, systems in Advanced Math almost always pair a linear equation with a quadratic. Rule: solve the linear equation for
The number of solutions depends on the resulting equation. If it reduces to a linear equation, there is exactly one solution. If it is quadratic, check the discriminant
- Positive → two solutions (line crosses the parabola twice)
- Zero → one solution (line is tangent to the parabola)
- Negative → no real solutions (line misses the parabola entirely)
Every algebraic solution is an intersection point on the graph. The SAT tests this connection from both directions: sometimes you solve algebraically and interpret graphically, sometimes you read a graph and confirm algebraically.
Concept Guides
5Solve systems involving linear and nonlinear equations (e.g., line with parabola).
Set the two y-expressions equal, solve the quadratic, and then recover both coordinates.
Interpret solutions as intersection points and determine how many solutions exist.
Use the discriminant to turn an intersection-count question into a quick solve for the parameter.
Match a system’s solutions to its graph.
Graph questions still come down to the same intersection points you would find algebraically.
Use substitution to reduce a system to one equation in one variable.
Pick the easiest variable to isolate, substitute, and solve the resulting one-variable equation.
Build a system from a context when two relationships are described.
Use perimeter and area to build a two-equation system for a rectangle.