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SAT Math Study Guide
For each subtopic: how to recognize it, how to solve it, and exactly what to type into Desmos — with each part explained. Try it live in the calculator below as you read.
Try typing any of the lines below into this calculator.
Algebra
Linear Equations
An equation where the variable only appears to the first power (no x², no x under a root). To solve, get all the variable terms on one side and all the plain numbers on the other.
Example: Solve 4x − 7 = 2x + 9
Subtract 2x from both sides, then add 7 to both sides: x = 8.
What to type into Desmos:
y_1=4x-7
y_2=2x+9
y_1= and y_2= just name each expression so Desmos can graph it and you can refer back to it.
Type the left side of your equation exactly as written.
Type the right side of your equation exactly as written.
Desmos draws both lines. Click the point where they cross — a bubble pops up showing its coordinates. The x-value of that point is your answer.
Functions
A function like f(x) is a rule: whatever number you put in for x, it tells you what comes out. "f(4)" means "plug in 4 for x and calculate."
Example: If f(x) = 3x − 5, find f(4).
f(4) = 3(4) − 5 = 7.
What to type into Desmos:
f(x)=3x-5
f(4)
Names the function f, using x as the input variable — type it exactly as the problem gives it.
The rule of the function — what happens to whatever x you plug in.
On a new line, type f( ) with the number you want to plug in. Desmos calculates the result automatically.
Desmos shows the value immediately next to the line — no graph-reading needed for this one.
Systems of Equations
Two equations sharing the same two variables. The solution is the one (x, y) pair that makes both equations true at once.
Example: 2x + y = 10 and x − y = 2. Find x + y.
Adding both equations eliminates y: 3x = 12, so x = 4, then y = 2. So x + y = 6.
What to type into Desmos:
y=10-2x
y=x-2
Rearrange each equation to solve for y first (move everything else to the other side), since Desmos graphs in terms of y.
First equation, rearranged: 2x + y = 10 becomes y = 10 − 2x.
Second equation, rearranged: x − y = 2 becomes y = x − 2.
Click the intersection point of the two lines — that point's (x, y) coordinates are the system's solution.
Slope
Slope measures steepness: how much y changes for each step in x. Given two points, slope = (change in y) ÷ (change in x).
Example: Find the slope between (1, 2) and (3, 8).
Slope = (8 − 2) / (3 − 1) = 6/2 = 3.
What to type into Desmos:
(8-2)/(3-1)
The change in y: second y-value minus first y-value.
Divide (the slope formula is a fraction).
The change in x: second x-value minus first x-value, in the same order as above.
Desmos computes the value instantly — no graphing needed for a pure arithmetic question like this.
Inequalities
Solved exactly like equations, except: if you multiply or divide both sides by a negative number, flip the inequality sign.
Example: Solve 2x + 3 < 11.
Subtract 3: 2x < 8. Divide by 2 (positive, so sign stays): x < 4.
What to type into Desmos:
y_1=2x+3
y_2=11
Graph both sides as if the inequality were an equation first.
The expression side of the inequality.
The number side of the inequality.
Find where the lines cross (x = 4). Since we need the expression to be less than 11, and its line is rising, the answer is everything to the left of that point: x < 4.
Word-Problem Modeling
Turning a real-world description into an equation. Look for a flat starting amount (the y-intercept, b) and a constant rate per unit (the slope, m): y = mx + b.
Example: $20 flat fee plus $0.10 per text — write an equation for total cost y after x texts.
Flat fee 20 is b. Rate 0.10 per text is m. So y = 0.10x + 20.
What to type into Desmos (to double-check your equation):
y=0.10x+20
The rate per unit (slope) — multiplied by x, the number of units.
The flat starting amount (y-intercept) — what you'd pay even with x = 0.
Check the graph crosses the y-axis at 20 — that confirms the flat fee is right, since that's the cost when x = 0 (no texts sent).