Systems of Equations

A system of equations is two (or more) equations that must be true at the same time. Solving means finding values for the variables that satisfy every equation in the system.

The solution is $x = 7$, $y = 3$ — both equations are satisfied.

Three solving methods. Pick whichever is fastest:

1. Substitution. Solve one equation for one variable, plug into the other.

• From $x + y = 10$: $x = 10 - y$.
• Plug into $x - y = 4$: $(10 - y) - y = 4 \Rightarrow 10 - 2y = 4 \Rightarrow y = 3$.
• Back to $x = 10 - 3 = 7$.

2. Elimination. Add or subtract equations to cancel a variable.

• Adding the two equations directly: $(x+y) + (x-y) = 10 + 4 \Rightarrow 2x = 14 \Rightarrow x = 7$.
• Then plug into either: $7 + y = 10 \Rightarrow y = 3$.

3. Graphing / inspection. Each linear equation is a line. The solution is where the lines cross. The SAT sometimes gives you a graph and asks you to read off the intersection.

Special cases:

• No solution: the lines are parallel (same slope, different y-intercepts). The system never has a common point.
• Infinitely many solutions: the lines are the same (same slope, same y-intercept). Every point on the line satisfies both equations.

Quick test for slopes-and-intercepts: rewrite both equations in $y = mx + b$ form. If the $m$'s differ, there's exactly one solution. If $m$'s match but $b$'s don't, no solution. If both match, infinite.

Word problems: translate two facts into two equations. "Tickets cost $12 for adults and $8 for students. They sold 100 tickets for $1,000." Two equations: $a + s = 100$ and $12a + 8s = 1000$.