SAT Math 2: Systems of Linear EquationsSAT Math 2: Systems of Linear Equations

Systems of Linear Equations

Goal: Solve systems by substitution, elimination, or graphing and decide how many solutions a system has.

THE SHORT VERSION

A system is two or more equations with the same variables. The solution is the point (or points) that satisfies all equations at once, which is where their graphs meet.

1 Methods

Substitution: solve one equation for a variable, then plug into the other. Best when a variable has coefficient 1. Elimination: add or subtract equations to cancel a variable. Best when coefficients match or are easy to match. Graphing (Desmos): type both equations and read the intersection point.

Example: 2x + y = 11 and x − y = 1. Add: 3x = 12, so x = 4, then y = 3. The solution is (4, 3).

2 How Many Solutions?

Graphs Slopes / intercepts Solutions

Cross Different slopes Exactly one

Parallel Same slope, different intercepts None

Same line Same slope and same intercept Infinitely many

For ax + by = c and dx + ey = f: no solution when a/d = b/e ≠ c/f; infinitely many when a/d = b/e = c/f.

Example: for what k do 3x + 2y = 6 and 6x + ky = 15 have no solution? Need 6/3 = k/2, so k = 4 (and 15/6 ≠ 2, so the lines are parallel, not identical).

Cross, parallel, or the same line.

3 Word Problems

Define one variable per unknown, write one equation per fact, and solve.

Example: 40 tickets are sold for $260. Adult tickets cost $8 and child tickets cost $5. Let a = adults and c = children: a + c = 40 and 8a + 5c = 260. Substitute c = 40 − a: 8a + 200 − 5a = 260, so 3a = 60 and a = 20, c = 20.

4 Linear and Quadratic Systems

Substitute the line into the quadratic and solve. The number of real solutions is the number of intersection points (0, 1, or 2). Example: y = x² and y = x + 2. Then x² − x − 2 = 0, so (x − 2)(x + 1) = 0 and x = 2 or −1; points (2, 4) and (−1, 1). 5 Check Yourself

Q: If x + y = 10 and x − y = 4, what is x? A: Add: 2x = 14, so x = 7.

Q: What is the value of 3x + 3y if x + y = 8? A: 3(x + y) = 24 (no need to solve for x and y).

6 Words to know

System of equations

Equations considered together.

Substitution

Replace a variable using another equation.

Elimination

Add or subtract equations to cancel a variable.

Inconsistent system

No solution (parallel lines).

Dependent system

Infinitely many solutions (same line).

SHORTCUT If the question asks for x + y or 2x + 2y, try adding or subtracting the equations directly. DESMOS Graph both equations and click the intersection.

SAT Math Guide 2 · original study notes