SAT Algebra: Every Question Type and Method

The digital SAT's Math section has four content domains, and Algebra is one of the biggest: about 35% of the section, or 13 to 15 of its 40 scored questions. Every Algebra question is about a line or a linear relationship, and the College Board splits it into five skills:

• Linear equations in one variable: solve for $x$, or turn a story into an equation.
• Linear functions: slope, intercepts, and rules like $f(x)=2x+8$.
• Linear equations in two variables: equations like $7s+11p=385$, and parallel and perpendicular lines.
• Systems of two linear equations: two equations, or two facts in a story, with the same two unknowns.
• Linear inequalities: limits like “at most $50”, and regions on a graph.

This SAT algebra review covers every algebra question type in six groups, with slope and intercepts split out of linear functions. The table below gives each one's method, and each section after it teaches the method, says when Desmos is faster, and gives you an SAT-style question to try. On most algebra questions, the math is the easy part: you earn the points by reading what each number means and what the question asks for.

How to answer an SAT algebra question

The same four moves work on every question in this guide:

1. Read the last line first. Is it asking for $x$, an expression like $x-4$, a slope, a count, or what a number means?
2. Name what each number means, with its unit: dollars per mile, books left, hours since the start.
3. Pick the faster route. When one or two steps get you there, work by hand. When the numbers are fractions or decimals, or two lines cross at messy numbers, Desmos is faster: it's built into the digital SAT for the whole Math section.
4. Answer the question, then check it against the equation or the story.

Each section below turns these moves into the steps for its question type. The Desmos course teaches the calculator itself.

If algebra questions are new to you, start with lesson 1.

Linear equation questions give you an equation, or a story to turn into one, and ask for a value. Often it isn't $x$: it's $x-4$, or $3x$, or the number of classes a student took. Others give an equation with a constant like $k$ and ask when it has no solution or infinitely many. Whatever the shape, find $x$, then answer the question.

How to answer linear equation questions

1. Read the last line and note what it asks for: $x$, an expression like $x-4$, or a constant.
2. Pick the route. With one or two clean steps, or an expression that repeats on both sides, solve by hand. With fractions, decimals or a long string of signs, type the equation into Desmos exactly as written and click its vertical line to read $x$: see solve SAT equations with Desmos.
3. Keep both sides equal: the number outside parentheses multiplies every term inside, combine only like terms, and do the same thing to both sides.
4. Answer the question: put your $x$ into whatever it asked for.

Solving linear equations

When an expression repeats, like $5-2x$ on both sides, keep it whole and solve for it instead of expanding. The lesson on how to solve SAT linear equations shows when each route is shorter. When the question asks for a multiple of one side, scale the whole equation instead: if $2x+3=11$, then $6x+9=33$.

Word problems

Say what your letter stands for, with its unit, before you write anything. Then give every number a job: a rate times an amount is one piece of a total, and a one-time fee is added once.

For $28 to register and $16 a class, $c$ classes cost $16c$. See SAT word problems.

No solution or infinitely many solutions

Simplify both sides and compare. Different $x$-coefficients give one solution. The same $x$-coefficients with different constants give no solution, and the same coefficients and constants give infinitely many. To find a constant, match the $x$-coefficients first, then check the constants. Algebra is usually faster than a graph here: see no solution and infinitely many solutions.

Mistakes to avoid with linear equations

The number outside the parentheses reaches every term, so $4(x-3)=4x-12$, and a minus sign in front of parentheses flips both signs inside: $-(x-9)=-x+9$. By hand or in Desmos, you end with $x$: reread the last line before you choose.

Slope and intercept questions give a line as an equation, two points, a table or a graph, and ask for its slope or where it crosses an axis. On the SAT, most come in a story and ask what the number means: a rate, a starting amount, the moment something runs out. A slope is a rate, and at an intercept the other variable is zero.

How to answer slope and intercept questions

1. Name the variables: what the input and the output measure, with units.
2. Find the slope: the change in output over the change in input, in the same order top and bottom. In an equation, solve for the output and read the number in front of the input. Its unit is output per input.
3. Find the intercept: set the other variable to $0$, by hand with clean numbers, or graph the line in Desmos and click the crossing when they're messy. A $y$-intercept, $(0,b)$, is the starting value; an $x$-intercept, $(a,0)$, is where the output reaches zero.
4. Read it back in the story, each coordinate in its own unit.

Slope and rate of change

Slope is $m=\frac{y_2-y_1}{x_2-x_1}$. Write one point above the other so each output stays with its own input, and subtract the same way top and bottom. In a story, the sign says whether the amount grows or falls: in $T(t)=84-1.6t$, the temperature drops $1.6$ degrees a minute. See SAT slope and rate of change.

Intercepts and starting values

The axis comes from the zero: $(7,0)$ is an $x$-intercept, whatever the $7$. Get $y$ alone before you read a constant, or set $x=0$: the $y$-intercept of $3y=6x-15$ is $-5$, not $-15$. With messy numbers, graph the line in Desmos and click the crossing, unless the question needs an exact value. See SAT linear intercepts, and Desmos points of interest SAT for reading the crossing.

Mistakes to avoid with slope and intercepts

Subtract in the same order, top and bottom: $\frac{8-2}{4-1}=2$. The $84$ is where the temperature starts, at $t=0$; the $-1.6$ is how fast it changes. And divide by $3$ before you read $b$.

Function questions hand you a line in disguise (a table, two points, a graph or a story) and ask for its rule. Others give the rule and ask for an output, or the input behind one. Model questions do both: build the rule from a story, then predict with it. Every linear rule comes down to two facts, a slope and one point.

How to answer linear function questions

1. Find the slope from two points: the change in output over the change in input.
2. Find $b$: put one point into $f(x)=mx+b$ and solve. A number is $b$ only when its input is $0$.
3. Check a second point in your rule, or in each choice. One matching point isn't enough. With awkward numbers, put the points in a Desmos table and graph the choices: the right line passes through every point.
4. Use the rule. Given an input, work forward. Given an output, set the rule equal to it and solve for the input.

Write and match linear functions

Table rows are points, and point-slope form, $y-y_1=m(x-x_1)$, is the same line with its slope and one point in view. With clean numbers, build the rule by hand; with awkward data and complete choices, graphing the points and the choices in Desmos can be less work: see Desmos tables and lists. See write SAT linear functions.

Inputs and outputs

In $f(a)=29$, $a$ is the input and $29$ is the output, so set the rule equal to $29$ and solve for $a$. Put a negative input in parentheses, and type a decimal equation like $18+6.5x=70$ into Desmos as it stands: see solve SAT equations with Desmos. See linear function inputs and outputs SAT.

Linear models

The same amount added at each step means linear: a start plus a rate times the number of steps. Then check the prediction against the story's limits. See SAT linear models.

Mistakes to avoid with linear functions

$29$ is an output, so the rule equals $29$. $-6$ is $b$ only if it happens at $h=0$: keep it as the point $(4,-6)$, so $T(h)=1.5(h-4)-6=1.5h-12$. And only the rate repeats: after $6$ days the drive has $145+22(6)=277$ books.

Two-variable questions give one equation in standard form, like $7s+11p=385$, that ties two amounts to one total, and ask what a number means, which pair works, or one amount when you know the other. Line questions add one word, parallel or perpendicular, that sets a second line's slope. The word sets the slope, and the point sets where the line goes.

How to answer line questions

1. Find the slope of the given line: solve for $y$, or use $-\frac{A}{B}$ for $Ax+By=C$.
2. Set the new slope. Parallel keeps it. Perpendicular flips the fraction and changes the sign, so the two slopes multiply to $-1$.
3. Use the point: put it into $y=mx+b$ to find $b$, or check that a choice passes through it and has the right slope.

Two-variable equations

Name what each letter counts. In $7s+11p=385$, the cost of $s$ standard and $p$ premium posters, $11$ is the price of one premium poster and $11p$ the price of all of them: the coefficient is for one, and the term is for all. One equation has many solution pairs; if you know one amount, plug it in and solve for the other. When an intercept means awkward division, type the equation into Desmos and click where it crosses the axis: see Desmos points of interest SAT. See linear equations in two variables.

Parallel and perpendicular lines

Compare slopes by hand: a graph can make nearly parallel lines look exactly parallel. A horizontal line, $y=c$, and a vertical line, $x=c$, are perpendicular to each other. See SAT parallel and perpendicular lines.

Mistakes to avoid with two-variable equations

$Ax+By=C$ hides its slope: solve for $y$ and $2x+3y=12$ has slope $-\frac23$. $11$ is the price of one poster, and $11p$ of all of them. And a perpendicular slope needs the flip and the sign change: $-2$.

System questions give two equations in the same two unknowns, or a story with two facts, and ask for the solution, one coordinate, a combination like $x+y$, or how many solutions there are. The solution is the point that makes both equations true, where the two lines cross. Answer the question, not the system: often you never need both values.

How to answer systems questions

1. Read what it asks for: $x$, $y$, or a combination like $x+y$.
2. Look at the coefficients. If adding or subtracting the equations (elimination) cancels a letter or builds what's asked, do that. If a letter is already alone, substitute. If elimination would take several steps, type both equations into Desmos and click the intersection: see solve systems of equations with Desmos.
3. Finish and check: work out only what's asked, and when you have the pair, put it back into both equations.

Solving systems

When you substitute, the whole expression goes in: with $y=2x+1$, $3x-y$ becomes $3x-(2x+1)$. When you scale an equation, multiply every term on both sides. See SAT systems of linear equations.

Word problems and how many solutions

Give each unknown a letter and turn each fact into its own equation, with every term measuring the same thing. Two lines cross once, never, or everywhere. If multiplying one equation's $x$- and $y$-coefficients by the same number gives the other's, the lines are parallel or the same line. Multiply its constant by that number too: if it matches the other constant, there are infinitely many solutions; if not, none. See SAT system of equations word problems.

For a team with $18$ games, no losses and $42$ points at $3$ a win and $1$ a draw, $3w+d$ counts points, so it can't equal $18$ games.

Mistakes to avoid with systems

Label the point $x=4$, $y=8$ before you answer. The minus sign reaches the whole expression: $3x-(2x+1)=x-1$. And a multiplier goes to both sides: $3(4x-y)=3(13)$.

Inequality questions ask how many, how long or how much before something hits a limit, or which point fits several conditions at once. One-variable inequalities solve like equations; two-variable ones are shaded regions on a graph. When only whole numbers make sense, don't round: check the whole numbers on each side of the boundary.

How to answer inequality questions

1. Write the inequality from the story: a start plus a rate times the amount, with $\le$ for “at most” or “without exceeding”, $\ge$ for “at least”, and $<$ or $>$ for “fewer than” or “more than”, which leave the boundary out.
2. Solve it like an equation, and flip the sign only when you multiply or divide both sides by a negative number.
3. Check the whole numbers on each side of the boundary: the answer is the one nearest it that still works, so the largest for a greatest value and the smallest for a least.

One-variable inequalities

A negative number somewhere isn't a reason to flip: only multiplying or dividing by one is. In a chain like $-3<2-4x\le10$, do each step to all three parts, and dividing by $-4$ flips both signs. $<$ and $>$ leave the boundary out; $\le$ and $\ge$ keep it. See SAT one-variable linear inequalities.

Two-variable inequalities

With two or more inequalities or several choices to test, Desmos is the faster route. Type every inequality as written and add the answer choices as points: a point counts only where all the shadings overlap, and a point on a dashed line doesn't count. Then check your point by hand in every inequality. See SAT two-variable linear inequalities, and Desmos inequalities for the Desmos steps.

Mistakes to avoid with inequalities

Subtracting $4$ keeps the sign: $-5x\le15$, and it flips only when you divide by $-5$, to $x\ge-3$. $9$ is past the boundary, so it breaks the limit. And $(1,-2)$ sits on the line $y=2x-4$, where $>$ leaves it out.