Lesson 1 of 5 · Algebra
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Linear equations in one variable are the foundation of the Algebra domain, which makes up about 35% of scored Math questions. They appear as quick early-module solves, as word problems you must translate, and as some of the hardest module 2 questions, where a constant such as must make an equation have no solution or infinitely many solutions.
Expect a few of these questions on every test, in both multiple-choice and student-produced response (SPR) format. There are five recurring forms:
| Form | Typical stem | Usual difficulty |
|---|---|---|
| Solve | “What value of is the solution to the given equation?” | Easy to medium |
| Solve for an expression | “If , what is the value of ?” |
3 questions on linear equations in one variable. Every option is explained once you answer.
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The first 3 of 11 cards for this topic. Sign in and finish the lesson to review them with spaced repetition.
First move for ?
| Easy to medium |
| Create from context | “Which equation can be used to find , the number of hours…?” | Easy to medium |
| Interpret a solution | “Which of the following is the best interpretation of in this context?” | Medium |
| Number of solutions and parameters | “In the given equation, is a constant. The equation has no solution. What is the value of ?” | Medium to hard |
The algebra is never long. What separates a 700 from an 800 is precision: distributing a negative sign correctly, clearing fractions without missing a term, and answering the quantity actually asked for.
For any linear equation, work in this order:
After simplifying, every linear equation looks like . Compare the coefficients of first and then the constants. You never need to finish solving.
| Condition | Solutions | What simplifying produces | Desmos picture (graph each side) |
|---|---|---|---|
| Exactly one | Two lines crossing once | ||
| and | None | A false statement, e.g., | |
| and | Infinitely many | A true statement, e.g., | One line (they coincide) |
A linear equation can never have exactly two solutions. If that choice appears, eliminate it.
When a question asks for something like or , look for the target expression inside the given equation before solving for . For example, and . Working with the whole expression is faster and avoids fractions.
Almost every context equation has the shape
The coefficient of the variable is the per-unit rate (dollars per hour, gallons per minute). The constant is the one-time or starting amount. Words such as drained, descends and spends make the rate negative. If only part of a quantity is charged, as in “after the first 200 messages”, the variable enters as .
0.4(x-3)=0.25x+1.8. Desmos draws a vertical line at the solution, and you click it to read . For a constant such as , Desmos offers a slider. Graph each side as its own equation and slide until the lines are parallel (no solution) or identical (infinitely many).10/3 directly. A long decimal must fill the space (3.333), and mixed numbers are not allowed. A positive answer can use up to 5 characters and a negative answer up to 6, including the minus sign.| Parallel lines |