Standard form: $Ax + By = C$

Slope-intercept form: $y = mx + b$

Point-slope form: $y - y_1 = m(x - x_1)$

Solving One-Step Equations

Use inverse operations to isolate the variable.

Example: $x + 7 = 12$ $x = 12 - 7 = 5$

Solving Two-Step Equations

Strategy: Undo addition/subtraction first, then undo multiplication/division.

Example: $3x - 5 = 16$

Step 1: Add 5 to both sides: $3x = 21$

Step 2: Divide both sides by 3: $x = 7$

Multi-Step Equations

When equations have variables on both sides, parentheses, or fractions:

Step-by-Step Strategy

1. Distribute any parentheses
2. Combine like terms on each side
3. Move variables to one side
4. Move constants to the other side
5. Divide to solve

Example: $4(2x - 3) = 5x + 6$ $8x - 12 = 5x + 6$ $3x = 18$ $x = 6$

Clearing Fractions

Multiply every term by the LCD (least common denominator).

Example: $\frac{x}{3} + \frac{x}{4} = 7$

LCD = 12, so multiply everything by 12: $4x + 3x = 84$ $7x = 84$ $x = 12$

Special Cases

No Solution

When simplifying leads to a false statement like $5 = 3$, there is no solution.

Example: $2(x + 1) = 2x + 5$ $2x + 2 = 2x + 5$ $2 = 5 \quad \text{(FALSE — no solution)}$

Infinitely Many Solutions

When simplifying leads to a true statement like $3 = 3$, there are infinitely many solutions.

Example: $3(x - 2) = 3x - 6$ $3x - 6 = 3x - 6$ $0 = 0 \quad \text{(TRUE — infinite solutions)}$

Linear Inequalities

Inequalities use $<$, $>$, $\leq$, $\geq$ instead of $=$.

The One Critical Rule

When you multiply or divide by a negative number, FLIP the inequality sign.

Example: $-3x > 12$ $x < -4 \quad \text{(sign flipped!)}$

Compound Inequalities

$-2 < 3x + 1 \leq 10$ Subtract 1 from all parts: $-3 < 3x \leq 9$ Divide all parts by 3: $-1 < x \leq 3$

SAT-Specific Question Types

Type 1: "What is the value of x?"

Straightforward solve — isolate the variable.

Type 2: "What is the value of an expression?"

Don't solve for x! Manipulate the equation to find the expression directly.

Example: If $3x + 5 = 17$, what is $6x + 10$? $6x + 10 = 2(3x + 5) = 2(17) = 34$

Type 3: "Which value is NOT a solution?"

Test each answer choice — the one that makes the inequality false is your answer.

Type 4: "For what value of $a$ does the equation have no solution?"

Set up the equation so the variable terms cancel and constants don't match.

Type 5: "For what value of $a$ does the equation have infinitely many solutions?"

Set up the equation so both sides are completely identical.

Word Problem Translation

Common SAT Mistakes

1. Forgetting to distribute the negative sign: $-(x - 3) = -x + 3$, NOT $-x - 3$
2. Not flipping the inequality sign when dividing by a negative
3. Solving for $x$ when the question asks for an expression like $2x + 1$
4. Arithmetic errors with fractions — always find the LCD first
5. Misreading "less than" — "$5$ less than $x$" means $x - 5$, NOT $5 - x$

Quick Reference: 5-Step SAT Strategy

1. Read the full question — what exactly are they asking for?
2. Simplify — distribute, combine like terms
3. Isolate — get the variable (or expression) alone
4. Check — plug your answer back in
5. Match — make sure your answer matches what was asked

Practice Tips

• On the SAT, you see roughly 6-8 linear equation questions
• Most can be solved in under 60 seconds with practice
• Watch for "shortcut" questions where you find an expression, not $x$
• If stuck, try plugging in answer choices (backsolving)

Find $y$: $y = 3(3) - 2 = 7$

Answer: $(3, 7)$ or $x = 3, y = 7$

Check: $2(3) + 7 = 13$ ✓

Divide by 0.10: $t = 120$

Answer: 120 text messages

SAT Tip: Set up word problems carefully - identify what the variable represents!

Step 3: Subtract $4x$ from both sides $2x - 5 = 5$

Step 4: Add 5 to both sides $2x = 10$

Step 5: Divide by 2 $x = 5$

Check: $3(2(5) - 4) + 7 = 3(6) + 7 = 25$ and $4(5) + 5 = 25$ ✓

Answer: $x = 5$

Step 2: Distribute $6x + 9 - 4x + 4 = 24$

Step 3: Combine like terms $2x + 13 = 24$

Step 4: Solve for $x$ $2x = 11$ $x = \frac{11}{2}$

Step 5: Find what the question asks: $10x - 5$ $10\left(\frac{11}{2}\right) - 5 = 55 - 5 = 50$

Answer: $10x - 5 = 50$

SAT Tip: The question asks for $10x - 5$, not $x$. Always read carefully!

Step 2: Subtract $2a$ from both sides $ax - 3 = 4x - 7$

Step 3: For NO solution, the $x$ coefficients must be equal but the constants must differ. Set the $x$ coefficients equal: $a = 4$

Step 4: Verify with $a = 4$: $4x - 3 = 4x - 7$ $-3 = -7 \quad \text{(FALSE — no solution ✓)}$

Answer: $a = 4$

Why this works: When $a = 4$, both sides have $4x$, but the constants ($-3$ vs $-7$) don't match, making the equation impossible.