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Introduction
This guide teaches the shortest reliable way to solve the Algebra questions on the digital SAT. Desmos handles most calculations. The lessons also cover the ideas that the calculator cannot decide for you.
How the guide is organized
The College Board divides Algebra into five skill labels. Those labels repeat many of the same solving methods. This guide instead groups questions by what you actually need to do:
1. Equations and intersectionsFind values, solve systems, and count solutions.
2. Lines, functions, and tablesMove between equations, graphs, points, and tables.
3. Building and interpreting modelsTranslate words and explain what parts of a model mean.
4. Inequalities and optimizationFind allowed values, regions, maximums, and minimums.
How to use each module
- Read the short lesson and general facts.
- Use the example buttons to build the setup in the calculator.
- Complete the easy, medium, and hard practice questions.
The calculator remains open on the right. Drag the divider to resize it, or close it when you need more reading space.
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Desmos basics
Use the simplest setup that matches the information given in the question.
| If the question gives you | Start with |
|---|---|
| An equation | Type it directly. If needed, graph each side separately. |
| Two equations | Graph both and click their intersection. |
| Points or a table | Add a table, then use y_1 ~ mx_1 + b. |
| An unknown constant | Add a slider for the constant, such as k or a. |
| Answer-choice equations | Graph the choices and compare their graphs. |
| An inequality | Type it directly and inspect the shaded solution region. |
Rules that prevent common mistakes
- Use function notation such as
f(x)=...when you need to evaluate the function, for examplef(5). You do not need it for every line. - Add sliders for unknown constants. Do not turn ordinary coordinate variables
xandyinto sliders. - A random value is useful only when the answer is guaranteed not to depend on that value.
- Always check restrictions from the context: counts are whole numbers, lengths are positive, and denominators cannot equal zero.
- Desmos can show a result. It cannot decide what a coefficient or point means in a word problem.
Try the main tools
Module 1
Equations and intersections
Use intersections to solve equations and systems. Then check what value the question actually asks for.
Use this module when
- You need to solve for a variable or evaluate an expression after solving.
- You are given two equations that must both be true.
- The question asks for zero, one, or infinitely many solutions.
- An unknown constant changes whether equations intersect.
Fast methods
One equation
Graph the left side as y=... and the right side as y=.... The intersection's x-coordinate solves the equation.
System
Graph both equations. The intersection point gives the values of both variables.
Unknown constant
Add a slider and adjust it until the graphs meet the stated condition.
Answer choices
Substitute or graph each choice when that is faster than solving symbolically.
General Facts
One intersectionOne solution
Parallel, separate linesNo solution
The same lineInfinitely many solutions
Denominator equals zeroThe expression is undefined
If the question asks for an expression such as $6x-1$, do not stop after finding $x$. Enter the full expression using the value you found.
Example: solve and evaluate
If $3x-8=7$, what is the value of $3x+8$?
Step 1: Type
y=3x-8.Step 2: Type
y=7 on the next line.Step 3: Click the intersection. Its x-coordinate is 5.
Step 4: Type
3(5)+8 to evaluate the requested expression.The intersection gives $x=5$. The requested answer is $3(5)+8=23$, not 5.
Example: number of solutions
How many solutions does $10(15x-9)=-15(6-10x)$ have?
Step 1: Type
y=10(15x-9).Step 2: Type
y=-15(6-10x).Step 3: Type
y=150x-90. Both sides simplify to this same line.The two graphs overlap completely, so the equation has infinitely many solutions.
Module 1
Equations and intersections: practice
In $(b-2)x=8$, $b$ is a constant. If the equation has no solution, what is the value of $b$?
The system $\frac12x+\frac13y=\frac16$ and $ax+y=c$ has infinitely many solutions, where $a$ and $c$ are constants. What is the value of $a$?
Module 2
Lines, functions, and tables
Use Desmos to move between equations, graphs, functions, points, and tables without repeatedly calculating slope and intercept by hand.
Use this module when
- You need an equation from points, a table, or a graph.
- You need a slope, x-intercept, or y-intercept.
- You need to evaluate a function.
- The question involves parallel, perpendicular, or shifted lines.
Fast methods
Points or table
Enter the points in a Desmos table. Then type y_1 ~ mx_1+b to find the line.
Function value
Define f(x)=..., then enter the requested value such as f(5).
Intercept
Graph the line and click where it crosses an axis.
Matching equation
Graph the answer choices and compare them with the given graph or points.
General Facts
SlopeChange in y for each one-unit change in x
x-interceptThe point where $y=0$
y-interceptThe point where $x=0$
Parallel linesEqual slopes
Perpendicular linesNegative reciprocal slopes
Example: equation from a table
A table contains the points $(3,8)$, $(6,18)$, and $(9,28)$, where $s$ is the input and $P$ is the output. Find the linear relationship between $s$ and $P$.
Step 1: Add a table. Enter 3, 6, and 9 in the $s_1$ column, then 8, 18, and 28 in the $P_1$ column.
Step 2: Type
P_1~ms_1+b below the table.Step 3: Desmos reports $m=\frac{10}{3}$ and $b=-2$. Type
y=(10/3)x-2.The relationship is $P=\frac{10}{3}s-2$.
Example: find an x-intercept
Line $k$ has slope 5 and y-intercept $(0,-35)$. What is the x-coordinate of the x-intercept of line $k$?
Step 1: Use the slope and y-intercept to type
y=5x-35.Step 2: Type
y=0 to show the x-axis as a second equation.Step 3: Click the intersection at $(7,0)$.
The x-coordinate of the x-intercept is 7.
Module 2
Lines, functions, and tables: practice
In the xy-plane, a line has a slope of 6 and passes through $(0,8)$. Which equation defines the line?
A linear relationship contains the points $(0.32,8)$, $(0.56,14)$, and $(0.68,17)$, where $d$ is distance in kilometers and $t$ is time in minutes. Which equation represents the relationship?
Line $h$ contains $(18,130)$, $(23,160)$, and $(26,178)$. Line $k$ is line $h$ translated down 5 units. What is the x-intercept of line $k$?
Module 3
Building and interpreting models
For many word problems, building the correct equation is the main task. Desmos becomes useful after the words have been translated correctly.
Build the model in this order
- Define each variable and write its unit.
- Identify totals, starting amounts, rates, and restrictions.
- Write one term for each part of the situation.
- Connect the terms with an equation, system, or inequality.
- Use Desmos to solve or inspect the model.
Total cost
(price)(quantity)Constant rate
(rate)(time)Starting amount plus change
start + rate * timePercent of an amount
decimal * amountTwo unknown groups
usually two equationsFirst amount plus additional items
first + rate(n - first count)Interpretation rules
CoefficientA per-unit amount when multiplied by a quantity
Constant termUsually an initial amount or fixed fee
Point $(x,y)$The two connected values described by the axes
SlopeHow much the output changes per input unit
Read the units before choosing an interpretation. For example, dollars per ticket is different from total dollars spent on tickets.
Example: build a system
Four shirts and two pairs of pants cost 86 dollars. Three shirts and five pairs of pants cost 166 dollars.
Step 1: Let $x$ be the shirt price and $y$ be the pants price. Type
4x+2y=86.Step 2: Type
3x+5y=166.Step 3: Click the intersection at $(7,29)$.
If $x$ is the shirt price and $y$ is the pants price, the equations are $4x+2y=86$ and $3x+5y=166$.
Example: interpret a linear model
A cargo-moving team uses $y=120-25x$ to estimate the number of tons of cargo remaining on a ship after working for $x$ hours. Interpret the x-intercept.
Step 1: Type
y=120-25x.Step 2: Type
y=0 because an x-intercept occurs when the output is zero.Step 3: Click the intersection at $(4.8,0)$.
The x-intercept means the team moved all the cargo in about 4.8 hours.
Module 3
Building and interpreting models: practice
In $x+y=75$, $x$ is the number of minutes spent running and $y$ is the number of minutes spent biking each day. What does 75 represent?
For a tape-dispenser business, $15{,}000=2.00x-4{,}500$. What is the best interpretation of $2.00x$?
A bin contained 24,000 bushels of corn. After an auger removed corn at a constant rate for 5 hours, 19,350 bushels remained. At the same rate, after how many total hours will 12,840 bushels remain?
Module 4
Inequalities and optimization
Inequalities describe every allowed value, not just one answer. Desmos shows these values as intervals or shaded regions.
Translate the comparison first
At least
≥At most / no more than
≤More than
>Less than
<Fast methods
One-variable inequality
Type it directly to see the allowed interval on the number line.
Two-variable inequality
Type it directly to see the shaded solution region.
System of inequalities
The overlap of all shaded regions contains the valid solutions.
Maximum or minimum
Inspect the boundary of the valid region, then apply context restrictions.
General Facts
- Solid boundaries belong to inequalities using $\le$ or $\ge$.
- Dashed boundaries do not belong to inequalities using $<$ or $>$.
- A point must satisfy every inequality in a system.
- Counts must be whole numbers. A graph value of 4.8 may mean a maximum of 4 items.
- Maximum and minimum questions often require checking a boundary or corner of the allowed region.
Example: system and maximum
A helicopter carries 120-pound packages $x$ and 100-pound packages $y$. It must carry at least 10 packages and at most 1,100 pounds. What is the maximum number of 120-pound packages?
Step 1: Type
x+y>=10 for at least 10 packages.Step 2: Type
120x+100y<=1100 for the weight limit.Step 3: Type
x>=0 because the number of 120-pound packages cannot be negative.Step 4: Type
y>=0 because the number of 100-pound packages cannot be negative.Step 5: Check whole-number points in the overlap. The greatest possible $x$ is 5.
The highest valid whole-number value of $x$ is 5, so the maximum number of 120-pound packages is 5.
Example: test a point
Which point is a solution to $y<-4x+4$?
Step 1: Type
y<-4x+4.Step 2: Type the first choice,
(-4,0).The point $(-4,0)$ lies in the shaded solution region, so it satisfies the inequality.
Module 4