SAT Algebra doesn't require heavy formula memorization, but a handful of relationships come up so often that knowing them cold saves real time on test day. This guide collects every one of them — slope, the three equation forms, parallel and perpendicular slopes, the no-solution/infinite-solutions shortcut, and the inequality flip rule — with worked examples showing exactly how each one gets used.
The slope formula
Given two points \(( x_1, y_1)\) and \((x_2, y_2)\), the slope between them is:
\(m = \dfrac{y_2 - y_1}{x_2 - x_1}\)
This single formula underlies linear functions, graphing, and any question that gives you two points or two rows of a table and asks for a rate of change. Keep the order of subtraction consistent in the numerator and denominator — it's easy to accidentally flip one and get the wrong sign. A slope of \(0\) describes a horizontal line, and an undefined slope (division by zero) describes a vertical line — the one straight-line graph that isn't a function, since it fails the vertical line test.
Slope-intercept, standard, and point-slope form
| Form | Formula | What you read off directly |
|---|---|---|
| Slope-intercept form | \(y = mx + b\) | Slope \(m\); y-intercept \(b\) |
| Standard form | \(Ax + By = C\) | Slope \(= -\dfrac{A}{B}\); y-intercept \(= \dfrac{C}{B}\) |
| Point-slope form | \(y - y_1 = m(x - x_1)\) | Builds an equation from one point and a slope |
Point-slope form is especially useful when you're given a slope and a single point but need a full equation — plug directly in without solving for \(b\) separately first. It's also the fastest route to an equation when a question describes a line "parallel to" or "perpendicular to" another line through a specific point, which is exactly where the next relationship comes in.
Parallel and perpendicular slopes
Two more relationships belong on this list, because the SAT tests them constantly — often without ever using the words "parallel" or "perpendicular" at all. A question might instead say a line "never intersects" another, or that two paths "cross at a right angle," and expect you to translate that into slope language on your own.
- Parallel lines always have equal slopes. If one line has slope \(m\), any line parallel to it also has slope \(m\).
- Perpendicular lines have slopes that are negative reciprocals of each other. If one line has slope \(m\), a line perpendicular to it has slope \(-\dfrac{1}{m}\).
Mini-example: A line passes through \((2, 1)\) and is perpendicular to the line \(y = -2x + 5\). What is the equation of this new line?
- Read the given slope directly from the equation: \(m = -2\).
- Take the negative reciprocal for the perpendicular slope: \(-\dfrac{1}{-2} = \dfrac{1}{2}\).
- Use point-slope form with the given point \((2, 1)\): \(y - 1 = \dfrac{1}{2}(x - 2)\)
- Distribute: \(y - 1 = \dfrac{1}{2}x - 1\)
- Add \(1\) to both sides: \(y = \dfrac{1}{2}x\)
Notice how little actual solving that required. Once you recognize the word "perpendicular," the slope is essentially handed to you, and point-slope form turns a single point into a full equation in two lines of work.
The no-solution / infinite-solutions condition
For an equation like \(ax + b = cx + d\), or a system of two equations, comparing the coefficients tells you the number of solutions without fully solving:
- If the coefficients on the variable differ (\(a \ne c\)), there's exactly one solution.
- If the coefficients match but the constants don't (\(a = c\), \(b \ne d\)), there's no solution.
- If both the coefficients and constants match (\(a = c\), \(b = d\)), there are infinitely many solutions.
The same logic extends to a system of two linear equations: write both equations in slope-intercept form, then compare slopes and y-intercepts using the table below.
| Slopes | Y-intercepts | Number of solutions |
|---|---|---|
| Different | Any | Exactly one |
| Same | Different | None |
| Same | Same | Infinitely many |
This table is really the parallel-lines rule from above wearing a different hat: "same slope, different intercept" is just a formal way of saying the two lines are parallel and therefore never meet.
The inequality flip rule
Not a formula so much as a rule you must apply automatically: multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. \(x < 5\) becomes \(-x > -5\) after multiplying both sides by \(-1\). Forgetting this single rule is one of the most common ways students lose otherwise-correct work on inequality questions — every other step can be flawless and the final answer still comes out backward.
The rule only applies to multiplication and division by a negative — adding or subtracting a negative number never flips anything. That distinction is worth keeping straight, since students who over-apply the rule end up flipping signs they shouldn't.
Worked example combining several formulas
A line passes through \((1, 4)\) and \((3, 10)\). Write its equation in slope-intercept form, then find its x-intercept.
- Find the slope: \(m = \dfrac{10 - 4}{3 - 1} = \dfrac{6}{2} = 3\)
- Use point-slope form with \((1, 4)\): \(y - 4 = 3(x - 1)\)
- Distribute: \(y - 4 = 3x - 3\)
- Add \(4\) to both sides: \(y = 3x + 1\)
- Find the x-intercept by setting \(y = 0\): \(0 = 3x + 1\)
- Solve: \(x = -\dfrac{1}{3}\)
This one example alone touches the slope formula, point-slope form, and slope-intercept form — a good reminder that these aren't five separate skills to drill in isolation, but one connected toolkit that most questions draw from more than once.
How these formulas carry into Advanced Math
These relationships aren't fenced off inside the Algebra domain — Advanced Math leans on the same logic in less obvious packaging, which is one more reason they're worth memorizing properly now rather than re-learning later:
- Quadratic and other nonlinear functions don't have a constant rate of change, but the average rate of change between two points on any curve still uses the identical slope formula, \(\dfrac{y_2 - y_1}{x_2 - x_1}\), applied to two points on the curve instead of a line.
- Point-slope reasoning reappears whenever an Advanced Math question asks you to build the equation of a line through a specific point using a given rate — the mechanics don't change just because the surrounding function got more complicated.
- The no-solution/infinite-solutions shortcut for linear systems is the direct ancestor of discriminant reasoning for quadratics — both let you count solutions by inspecting coefficients instead of solving the whole equation.
Treat this page as a foundation, not a finish line: the faster slope, intercepts, and solution-counting become automatic here, the less mental overhead you'll spend re-deriving them once questions move into quadratics and other nonlinear territory.
Quick-reference formula table
| Concept | Formula |
|---|---|
| Slope from two points | \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\) |
| Slope-intercept form | \(y = mx + b\) |
| Point-slope form | \(y - y_1 = m(x - x_1)\) |
| Slope from standard form | \(m = -\dfrac{A}{B}\) |
| Parallel slopes | Equal: \(m_1 = m_2\) |
| Perpendicular slopes | Negative reciprocals: \(m_1 = -\dfrac{1}{m_2}\) |
| Inequality flip rule | Multiply/divide by negative \(\Rightarrow\) flip sign |
Tip: don't just memorize these — derive each one once by hand. Formulas you've derived stick under pressure far better than ones you've only read.
Frequently asked questions
Is it enough to just memorize this list?
Memorizing gets you halfway there. The rest comes from applying each formula inside real problems, including the messier ones covered in Linear Equations in One Variable: SAT Math Guide and beyond, until recognizing when to use each formula becomes automatic.
Does the SAT give a formula sheet with these on it?
The reference sheet provided on the digital SAT covers geometry formulas (area, volume, angle relationships) — it does not include the Algebra formulas on this page, so these genuinely need to be memorized.
How is a "negative reciprocal" different from just a negative slope?
A negative reciprocal flips the fraction and changes the sign at the same time — the negative reciprocal of \(\dfrac{2}{3}\) is \(-\dfrac{3}{2}\), not \(-\dfrac{2}{3}\). Confusing the two is the most common error on perpendicular-slope questions, so it's worth double-checking that you did both steps, not just the sign.
Do these same formulas apply to Advanced Math questions?
The exact forms above describe linear relationships specifically, which is an Algebra-domain concept, but the underlying logic behind them — rate of change between two points, building an equation from limited information, counting solutions without fully solving — carries directly into Advanced Math topics like quadratics and other nonlinear functions.
What's the best way to drill these until they're automatic?
Timed practice beats flashcards for this. Work through Easy, Medium, and Hard sets on SAT Math practice sets and pay attention to which formula each question is really testing — after a few sets, the pattern recognition becomes instinctive. See Algebra Practice Questions: Free SAT Math Sets by Difficulty for a full study plan.