9.1 Linear Equations in One Variable
Key Takeaways
- Isolate x by distributing first, combining like terms, then undoing addition or subtraction and undoing multiplication or division on both sides.
- Clear an equation with fractions by multiplying every term by the least common denominator so the arithmetic uses integers.
- If the x terms cancel, a true leftover such as 0 = 0 means infinitely many solutions; a false leftover such as 5 = -1 means no solution.
- About 30% of PSAT 8/9 Math is in context: write the ticket, age, or temperature equation on paper before opening Desmos.
- Algebra is about 42.5% of operational Math (16–18 questions); a calculator is allowed on every Math item, but Desmos checks a setup—it does not invent the equation.
Algebra is the largest Math domain on the PSAT 8/9. College Board's Assessment Framework for the Digital SAT Suite puts Algebra at about 42.5% of operational Math, 16–18 questions. This independent OpenExamPrep chapter teaches three of the five Algebra skill/knowledge testing points: linear equations in one variable, linear equations in two variables, and linear functions. Systems and inequalities come in the next chapter. OpenExamPrep is not a College Board partner and does not claim official approval, review, partnership, or exact equivalence with College Board materials. Use the Student Guide, Bluebook practice, and this guide together.
On this test, every Math question allows a calculator. Bluebook includes Desmos (scientific and graphing). A handheld non-CAS calculator is allowed; CAS devices are not. About 30% of Math items are in context—science, social studies, or a real-world sentence. About 25% of Math is student-produced response (SPR): you type a number instead of picking A–D. Fluency with isolating x still matters, because neither Desmos nor a four-function calculator can invent the equation from a sentence. Calculator rules live in 3.1 Devices, What to Bring, and Calculator Policy. SPR entry rules live in 15.1 Student-Produced Response Questions.
What a linear equation in one variable looks like
A linear equation in one variable has the unknown to the first power only. Typical looks:
- 3x + 11 = 32
- 4(x - 3) + 5 = 2x + 9
- x/3 + x/6 = 5
Not this chapter, and not a PSAT 8/9 Algebra item: x squared equals 9, 1/x = 4, or a square-root equation. College Board does not assess rational or radical equations on PSAT 8/9, and this test does not assess trigonometry or circle theorems. Stay on first-power equations until later Advanced Math chapters that stay inside the published PSAT 8/9 list.
The job is to isolate x: undo operations so x sits alone, keeping both sides equal at every step.
| Move | Why it is legal |
|---|---|
| Add or subtract the same number on both sides | Equality is preserved |
| Multiply or divide both sides by the same nonzero number | Equality is preserved |
| Distribute a(b + c) = ab + ac | Multiplication over addition |
| Combine like terms | 5x and 2x are the same kind of term |
Worked example: isolate x
Solve 3x + 11 = 32.
Subtract 11 from both sides: 3x = 21. Divide both sides by 3: x = 7.
Check in the original: 3(7) + 11 = 21 + 11 = 32. Both sides match.
If this were SPR, you would type 7. Do not type x=7. The box wants the number. Mixed numbers must be improper fractions or decimals. Do not enter a percent sign, a comma, or a dollar sign. A negative answer needs the minus sign; College Board allows up to 5 characters for a positive entry and 6 with the negative sign.
Worked example: distribute, then like terms
Solve 4(x - 3) + 5 = 2x + 9.
Distribute the 4: 4x - 12 + 5 = 2x + 9. Combine -12 + 5: 4x - 7 = 2x + 9. Subtract 2x from both sides: 2x - 7 = 9. Add 7: 2x = 16. Divide by 2: x = 8.
Check the left side: 4(8 - 3) + 5 = 4(5) + 5 = 20 + 5 = 25. Check the right side: 2(8) + 9 = 16 + 9 = 25.
A common grade 8 error is writing 4(x - 3) as 4x - 3. The 4 must hit both terms inside the parentheses. If your check fails, look there first.
Worked example: a negative distribute
Solve 3(2x - 5) - 4 = 5x + 1.
Distribute 3: 6x - 15 - 4 = 5x + 1. Combine -15 - 4: 6x - 19 = 5x + 1. Subtract 5x: x - 19 = 1. Add 19: x = 20.
Check left: 3(2 · 20 - 5) - 4 = 3(40 - 5) - 4 = 3(35) - 4 = 105 - 4 = 101. Check right: 5(20) + 1 = 100 + 1 = 101.
Worked example: clear denominators
Fractions slow you down if you move one piece at a time. Multiply every term by the least common denominator (LCD) so the new equation uses integers.
Solve x/3 + x/6 = 5.
The LCD of 3 and 6 is 6. Multiply every term by 6:
6 · (x/3) + 6 · (x/6) = 6 · 5
2x + x = 30
3x = 30
x = 10
Check: 10/3 + 10/6 = 10/3 + 5/3 = 15/3 = 5.
Second fraction example: 2x/5 - 3 = 1. Add 3: 2x/5 = 4. Multiply both sides by 5: 2x = 20. x = 10. Check: 20/5 - 3 = 4 - 3 = 1.
Third: x/4 + 5 = x/2 - 1. LCD 4. Multiply through by 4:
4 · (x/4) + 4 · 5 = 4 · (x/2) - 4 · 1
x + 20 = 2x - 4
20 + 4 = 2x - x
x = 24
Check: 24/4 + 5 = 6 + 5 = 11, and 24/2 - 1 = 12 - 1 = 11.
One solution, no solution, or an identity
After you simplify, three endings appear. Read the leftover line; do not invent an x that is not there.
| Ending after simplifying | Name | What it means |
|---|---|---|
| x equals a number | One solution | Exactly one value works |
| A true number sentence with no x left, such as 0 = 0 | Identity | Every real number works (infinitely many solutions) |
| A false number sentence, such as 5 = -1 | Contradiction | No real number works |
One solution. 5x - 2 = 3x + 10. Subtract 3x: 2x - 2 = 10. Add 2: 2x = 12. x = 6. Check: 5(6) - 2 = 30 - 2 = 28, and 3(6) + 10 = 18 + 10 = 28.
Identity. 2(x + 4) = 2x + 8. Left side: 2x + 8. Both sides are identical, so 0 = 0. Infinitely many solutions.
No solution. 3x + 5 = 3x - 1. Subtract 3x: 5 = -1, which is never true. No solution.
College Board's later SAT Suite Algebra list names conditions for no solution, a unique solution, or infinitely many solutions more explicitly. The published PSAT 8/9 one-variable description stresses creating, interpreting, and fluently solving. If the variable terms cancel on this test, you still have to read the leftover numbers: true means all x, false means none. That is ordinary grade 8 algebra, and it is how you avoid bubbling a fake x = 0.
Word problems: tickets, age, temperature
In-context items are still one-variable equations. Name the unknown, write one equation, then isolate.
Tickets. Student tickets cost $9 and adult tickets cost $14. A family buys 2 adult tickets and s student tickets and spends $55.
14(2) + 9s = 55
28 + 9s = 55
9s = 27
s = 3
Check: 28 + 27 = 55. Optional Desmos check: graph y = 9x + 28 and y = 55 and read the intersection x = 3. That graph only works because you already wrote 14(2) + 9s = 55. Desmos does not know that 2 is the adult count unless you put it in.
Age. Lina is 3 years younger than twice Omar's age. Lina is 15. Let o be Omar's age in years.
15 = 2o - 3
18 = 2o
o = 9
Check: twice 9 is 18; 18 minus 3 is 15.
Temperature. A freezer starts at -8°C and warms 2°C each hour. After h hours the temperature is 6°C.
-8 + 2h = 6
2h = 14
h = 7
Check: -8 + 14 = 6.
Desmos as a check, not a setup
Use the embedded graphing calculator to verify. Do not use it as a substitute for translating the sentence.
- Graph y = (left side) and y = (right side). The x-coordinate of the intersection is the solution of a one-variable equation.
- Recompute 3(40 - 5) - 4 on the scientific keypad after you already isolated x = 20.
- For 4(x - 3) + 5 = 2x + 9, plot y = 4(x - 3) + 5 and y = 2x + 9 and confirm they meet at x = 8.
If you type the wrong expression, Desmos will confirm the wrong answer with perfect confidence. Graphing habits return in 14.3 Reference Sheet, Desmos, and What This Test Does Not Include.
Test-day habits for this skill
Copy the equation onto the school scratch paper (write your name; do not tear the packet; do not bring your own paper). Circle the operation you are undoing. Substitute the result into the original equation, not into a half-simplified line. On multiple choice, plugging the four numbers is a legal backup if algebra stalls. On SPR, a check is more important because no options catch a sign error.
Math modules run easiest to hardest, but Algebra appears in both modules. A one-variable solve can be an early gift or a late in-context SPR. Bank the clean isolate-x items; save extra seconds for a story with a hidden like-term step.
What is the solution of 5x - 9 = 2x + 12?
What is the solution of 6(x + 2) = 4x + 20?
After simplifying 4(x - 1) + 2 = 4x - 2, which description of the solution set is correct?