Linear equation questions are among the most common math problems on the SAT. They ask you to solve for one variable, set up an equation from a word problem, or interpret what a variable or coefficient means.
A linear equation has the form: ax + b = c (no squared terms, no fractions with variables in the denominator).
Goal: Get the variable alone on one side. Rule: Whatever you do to one side, do to the other.
Example 1 (Simple): > 3x + 5 = 20 > Subtract 5: 3x = 15 > Divide by 3: x = 5
Example 2 (Variables on both sides): > 4x − 3 = 2x + 9 > Subtract 2x: 2x − 3 = 9 > Add 3: 2x = 12 > Divide by 2: x = 6
Example 3 (Distributive property): > 2(3x + 4) = 26 > Distribute: 6x + 8 = 26 > Subtract 8: 6x = 18 > Divide by 6: x = 3
This is where most students struggle. The key is translating English into math.
| English phrase | Math symbol | |---|---| | "is," "equals," "results in" | = | | "sum of," "more than," "increased by" | + | | "difference," "less than," "decreased by" | − | | "product," "times," "of" | × | | "divided by," "per," "ratio" | ÷ | | "a number" or "the unknown" | x (or any variable) |
Example: "A store sells notebooks for $3 each and pens for $1.50 each. A student spent $18 buying notebooks and pens. She bought twice as many pens as notebooks. How many notebooks did she buy?"
The SAT often asks: "What does the value of [number] represent in the equation?"
> A parking garage charges a $5 entry fee plus $2 per hour. Total cost C = 2h + 5. > - The 2 represents the rate of change: $2 per additional hour > - The 5 represents the initial/starting fee (when h = 0)
Real-world example: "A gym membership costs $30 to join and $20 per month. The equation T = 20m + 30 represents total cost T after m months. What does 20 represent?" → The monthly cost (cost for each additional month). What does 30 represent? → The one-time joining fee.
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Quiz Questions:
Q1. Solve: 5x − 8 = 2x + 10
A) x = 2 B) x = 6 C) x = 18/7 D) x = 4
Answer: B — Subtract 2x from both sides: 3x − 8 = 10. Add 8: 3x = 18. Divide by 3: x = 6.
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Q2. A taxi charges a flat fee of $4 plus $2.50 per mile. The equation C = 2.50m + 4 represents the total cost C for m miles. What does 4 represent?
A) The cost per mile B) The total number of miles C) The flat fee charged at the start of every ride D) The distance to the passenger's destination
Answer: C — In the equation C = 2.50m + 4, the 4 is the constant — the value of C when m = 0. This represents the flat fee paid at the start of every ride, before any miles are driven.
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Q3. A student earns $12 per hour at her part-time job. She has already saved $85 and wants to save a total of $265. Which equation represents this situation, where h is the number of additional hours she needs to work?
A) 12h = 265 B) 12h + 85 = 265 C) 12h − 85 = 265 D) 85h + 12 = 265
Answer: B — She starts with $85 and earns $12 per additional hour. Total savings = existing savings + new earnings. Setting that equal to $265: 12h + 85 = 265.
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Q4. Solve: 3(2x − 4) = 3x + 9
A) x = 7 B) x = 3 C) x = 21 D) x = 5
Answer: A — Distribute: 6x − 12 = 3x + 9. Subtract 3x: 3x − 12 = 9. Add 12: 3x = 21. Divide by 3: x = 7.
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Q5. An equation is written as y = −3x + 15. What does the −3 represent?
A) The starting value of y when x = 0 B) The value of x when y = 0 C) The rate at which y decreases for each 1-unit increase in x D) The maximum value of y
Answer: C — In the slope-intercept form y = mx + b, the coefficient of x is the slope, which represents the rate of change. A slope of −3 means y decreases by 3 for every 1-unit increase in x. The 15 (not −3) is the y-intercept (starting value).