Math Formulas/Equations

What You Need to Know

SAT Math rewards two things: (1) knowing the core formulas and (2) being able to build and solve equations quickly and cleanly. This sheet is the “night-before” toolbox for the most-tested equation types and the formulas you’ll actually use.

The big idea

Most SAT problems reduce to one of these moves:

• Translate words to an equation (define a variable, write a relationship, solve).
• Rewrite an expression (factor, expand, combine like terms, use exponent rules).
• Solve for an unknown (linear, quadratic, system, inequality, rational, radical, absolute value).
• Plug into a formula (slope, distance, area/volume, circle, percent, interest).

Core equation forms you must recognize

• Linear (one variable): $ax+b=c$
• Linear (two variables): $y=mx+b$
• Standard form: $Ax+By=C$
• Quadratic: $ax^2+bx+c=0$ or $y=ax^2+bx+c$
• Exponential (growth/decay): $A(t)=A_0(1+r)^t$
• Direct variation: $y=kx$; **inverse variation:** $y=\frac{k}{x}$

Critical reminder: Any time you square, cross-multiply, or multiply both sides by a variable expression, you must check for extraneous solutions and domain restrictions.

Step-by-Step Breakdown

1) Solving a linear equation (fast + safe)

1. Distribute if needed: $a(b+c)=ab+ac$
2. Combine like terms on each side.
3. Move variables to one side, constants to the other (use add/subtract).
4. Divide to isolate the variable.

Mini example: Solve $3(2x-5)=x+7$

• Distribute: $6x-15=x+7$
• Subtract $x$: $5x-15=7$
• Add $15$: $5x=22$
• Divide: $x=\frac{22}{5}$

2) Solving a system of linear equations

Method A: Elimination (usually fastest)

1. Write both in $Ax+By=C$ form if helpful.
2. Multiply one/both equations so a variable coefficient matches.
3. Add/subtract equations to eliminate one variable.
4. Solve for the remaining variable.
5. Back-substitute to find the other variable.

Mini example:
$\begin{cases}2x+y=11\\3x-y=4\end{cases}$
Add equations: $5x=15 \Rightarrow x=3$
Back-substitute: $2(3)+y=11 \Rightarrow y=5$

Method B: Substitution (best when one variable is isolated)

1. Solve one equation for $x$ or $y$.
2. Substitute into the other.
3. Solve, then back-substitute.

3) Solving a quadratic

Option A: Factor (if it factors nicely)

1. Set to zero: $ax^2+bx+c=0$
2. Factor: $(px+q)(rx+s)=0$
3. Zero-product rule: $px+q=0$ or $rx+s=0$

Option B: Quadratic formula (always works)

1. Identify $a,b,c$ in $ax^2+bx+c=0$.
2. Use $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
3. Simplify; if asked for number of solutions, check discriminant $\Delta=b^2-4ac$.

Decision point:

• If factoring is obvious, factor.
• If not, go straight to the quadratic formula.

4) Rational equations (fractions with variables)

1. Find the LCD (least common denominator).
2. Multiply every term by the LCD.
3. Solve the resulting equation.
4. Check solutions in the original (denominators cannot be $0$).

Mini example: Solve $\frac{x}{x-2}=3$

• Multiply by $x-2$: $x=3(x-2)$
• Solve: $x=3x-6 \Rightarrow -2x=-6 \Rightarrow x=3$
• Check: $x\neq 2$, so $x=3$ is valid.

5) Radical equations (variables under a square root)

1. Isolate the radical.
2. Square both sides.
3. Solve.
4. Check (squaring can create extraneous solutions).

6) Absolute value equations and inequalities

Key idea: $|A|$ measures distance from $0$.

• Equation: $|A|=k$ (with $k\ge 0$) becomes $A=k$ or $A=-k$.
• Inequality: $|A|<k$ becomes $-k<A<k$.
• Inequality: $|A|>k$ becomes $A>k$ or $A<-k$.

7) Inequalities (don’t miss the flip)

1. Solve like an equation.
2. Flip the inequality sign when multiplying/dividing by a negative.

Key Formulas, Rules & Facts

Algebra essentials (manipulation + structure)

Exponents & radicals

Linear functions & coordinate geometry

Quadratics (graphs, roots, vertex)

Ratios, proportions, percent

Geometry formulas that show up inside equations

Circle in the coordinate plane

Right-triangle trig (equations built from ratios)

Examples & Applications

Example 1: Build an equation from words (percent)

A jacket is discounted $20\%$ from original price $p$, then the discounted price is $48$. Find $p$.

• Discounted price: $p-0.20p=0.80p$
• Equation: $0.80p=48$
• Solve: $p=\frac{48}{0.80}=60$
  Pattern: “After a $k\%$ decrease” means multiply by $1-k$ (as a decimal).

Example 2: System from a context (two unknowns)

You buy $3$ coffees and $2$ sandwiches for $\$19$, and $2$ coffees and $3$ sandwiches for $\$20$. Let coffee cost $c$ and sandwich cost $s$.

• Equations: $3c+2s=19$ and $2c+3s=20$
• Eliminate: multiply first by $3$ and second by $2$:
  • $9c+6s=57$
  • $4c+6s=40$
• Subtract: $5c=17 \Rightarrow c=\frac{17}{5}$
• Back-substitute: $3\left(\frac{17}{5}\right)+2s=19 \Rightarrow 2s=\frac{44}{5} \Rightarrow s=\frac{22}{5}$
  Pattern: Set up two equations from two purchases; elimination is usually clean.

Example 3: Quadratic (factoring vs formula)

Solve $x^2-5x-14=0$.

• Factor: find numbers that multiply to $-14$ and add to $-5$: $-7$ and $2$.
• $(x-7)(x+2)=0$
• Solutions: $x=7$ or $x=-2$
  Variation: If it doesn’t factor quickly, use $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.

Example 4: Radical equation (extraneous trap)

Solve $\sqrt{x+5}=x-1$.

• Domain: need $x-1\ge 0 \Rightarrow x\ge 1$
• Square: $x+5=(x-1)^2=x^2-2x+1$
• Rearrange: $0=x^2-3x-4$
• Factor: $(x-4)(x+1)=0 \Rightarrow x=4$ or $x=-1$
• Check domain and original:
  • $x=4$ works: $\sqrt{9}=3$
  • $x=-1$ fails domain and original
    Answer: $x=4$.

Common Mistakes & Traps

1. Forgetting to distribute a negative

   • Wrong: turning $-(x-3)$ into $-x-3$.
   • Right: $-(x-3)=-x+3$.
   • Fix: treat the negative as multiplying everything inside.

2. Not flipping an inequality when multiplying/dividing by a negative

   • If you multiply by $-2$, $x<3$ becomes $-2x>-6$.
   • Fix: say out loud: “negative means flip.”

3. Cross-multiplying when you shouldn’t (or ignoring zeros)

   • In $\frac{a}{b}=\frac{c}{d}$, you need $b\neq 0$ and $d\neq 0$.
   • Fix: note denominator restrictions first.

4. Extraneous solutions from squaring or clearing denominators

   • Squaring both sides can add solutions.
   • Rational equations can “allow” a value that makes a denominator $0$.
   • Fix: always plug solutions back into the original equation.

5. Mixing up slope formula order

   • Wrong: $\frac{y_2-y_1}{x_1-x_2}$ (only one difference reversed).
   • Fix: keep consistent: $\frac{y_2-y_1}{x_2-x_1}$.

6. Assuming $\sqrt{a^2}=a$ (missing absolute value)

   • Truth: $\sqrt{a^2}=|a|$.
   • Fix: if you simplify a squared expression under a root, consider both signs.

7. Misreading intercepts and parameters

   • In $y=mx+b$, $b$ is the $y$-intercept (not $x$-intercept).
   • In $(x-h)^2+(y-k)^2=r^2$, center is $(h,k)$ (signs matter).
   • Fix: memorize “opposite sign” behavior: $x-h$ means center at $h$.

8. Dropping parentheses in substitution

   • If $y=2x-3$ and you plug into $x+y=10$, write $x+(2x-3)=10$.
   • Fix: always wrap substituted expressions in parentheses.

Memory Aids & Quick Tricks

Quick Review Checklist

• You can rewrite between $y=mx+b$, $y-y_1=m(x-x_1)$, and $Ax+By=C$.
• You know slope, distance, midpoint: $m=\frac{y_2-y_1}{x_2-x_1}$, $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.
• You can solve systems by elimination (and choose smart multiples).
• You can solve quadratics by factoring or $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
• You automatically check: denominator $\neq 0$, radicand constraints, and extraneous solutions.
• You handle $|A|=k$ as $A=k$ or $A=-k$ and absolute value inequalities as “between” or “outside.”
• You never forget to flip the inequality when multiplying/dividing by a negative.
• You can set up percent equations using $\text{part}=\text{percent}\cdot\text{whole}$.

You’ve got the tools—now it’s just pattern recognition and clean execution.