Linear Equations and Inequalities (SAT)

One-Variable Linear Equations

SAT Format: You'll see equations to solve or word problems requiring setup.

Example: Solve for $x$: $3(x - 2) = 15$

Steps:

1. Distribute: $3x - 6 = 15$
2. Add 6: $3x = 21$
3. Divide by 3: $x = 7$

Linear Inequalities

Key Difference: When multiplying/dividing by a negative, flip the inequality sign!

Example: $-2x + 5 > 11$

• Subtract 5: $-2x > 6$
• Divide by -2 (flip sign!): $x < -3$

Systems of Linear Equations

Two methods:

Substitution

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

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

Elimination

$\begin{cases} 2x + 3y = 12 \\ 4x - 3y = 6 \end{cases}$

Add equations: $6x = 18 \quad \Rightarrow \quad x = 3$

SAT Tips

• Check your answer by substituting back
• Word problems: Identify the variable, write equation, solve
• Watch for tricks: $2x = 10$ asks for $x$, not $2x$