Systems of Linear Equations (SAT)

What is a System of Equations?

Two or more equations with the same variables

Goal: Find values that satisfy ALL equations simultaneously

SAT Solution Methods

Method 1: Substitution

Best when: One equation is already solved for a variable

Steps:

1. Solve one equation for one variable
2. Substitute into the other equation
3. Solve for remaining variable
4. Back-substitute to find other variable

Example: $\begin{cases} y = 2x + 1 \\ 3x + y = 11 \end{cases}$

Substitute $y = 2x + 1$ into second equation: $3x + (2x + 1) = 11$ $5x = 10$ $x = 2, \quad y = 5$

Method 2: Elimination (Addition/Subtraction)

Best when: Coefficients line up nicely

Steps:

1. Multiply equations to get matching coefficients
2. Add or subtract to eliminate one variable
3. Solve for remaining variable
4. Substitute back

Example: $\begin{cases} 2x + 3y = 12 \\ 2x - y = 4 \end{cases}$

Subtract equations: $4y = 8 \quad \Rightarrow \quad y = 2$ $x = 3$

Method 3: Graphing

Best when: Answer choices show intersection points

Key insight: Solution = where lines cross

For SAT:

• Might show you the graph
• Ask "where do they intersect?"
• Answer = $(x, y)$ coordinate

Special Cases

No Solution (Parallel Lines)

Same slope, different y-intercepts

$\begin{cases} y = 2x + 3 \\ y = 2x - 1 \end{cases}$

Lines never cross!

Infinitely Many Solutions (Same Line)

Equations are multiples of each other

$\begin{cases} 2x + 4y = 8 \\ x + 2y = 4 \end{cases}$

Second equation × 2 = First equation!

SAT Question Types

Type 1: Direct Solve

"What is the solution to the system?"

Straightforward - use any method

Type 2: Value of Expression

"What is $x + y$?"

Don't need individual values - look for shortcut!

Example: $\begin{cases} 3x + 2y = 10 \\ x + y = ? \end{cases}$

Sometimes you can add/subtract equations directly

Type 3: Which Point Satisfies Both?

"Which ordered pair $(x,y)$ is a solution?"

SAT Trick: Plug in answer choices!

Type 4: Number of Solutions

"How many solutions does the system have?"

• Different slopes → 1 solution
• Same slope, different intercepts → 0 solutions
• Same line → Infinite solutions

SAT Strategies

Calculator Tip

Your calculator can solve systems!

1. Graph both equations
2. Find intersection point
3. Verify with answer choices

Check Your Answer

Plug $(x, y)$ back into BOTH equations

Work Backwards

If given answer choices, test them!

Look for Shortcuts

Sometimes adding equations gives you what you need

Common SAT Traps

Trap 1: Only solving for one variable

Question asks for $x$, you find $y$ → keep going!

Trap 2: Arithmetic errors

Always check by substituting back

Trap 3: Confusing $x$ and $y$

Answer choices like $(3, 5)$ vs $(5, 3)$

Trap 4: Forgetting to simplify

May need to reduce fractions or combine terms

SAT Tips

• Substitution when equation already solved
• Elimination when coefficients match up
• Graphing when you have a calculator
• Plug in answers when given choices
• Check your work - takes 10 seconds, saves points!