🎯 Learning Objectives
- How a linear equation is built and what makes it 'linear'.
- Solve linear equations by simplifying and applying equivalent operations.
- How to spot equations with no solution vs. infinite solutions.
- Choose an efficient method and check that the solution satisfies the original equation.
- Practice questions with worked solutions.
1. What Is a Linear Equation?
A linear equation is an equation in which every variable appears only to the first power. There are no squares, no square roots, no variables in denominators, and no variables hiding inside absolute value bars.
A linear equation ax + b = 0 with a ≠ 0 has one solution. After simplification, an equation may instead be an identity or a contradiction. In two variables, ax + by = c graphs as a straight line when a and b are not both zero: that's where the name comes from.
Standard one-variable form
a · x + b = c
Slope-intercept form (two variables)
y = m · x + b
Linear or Not? Quick Recognition Table
Equation | Linear? | Why |
3x − 7 = 11 | ✅ Yes | x is to the 1st power |
5(x + 2) = 4x − 1 | ✅ Yes | Distributes to a linear form |
x² + 2x = 9 | ❌ No | x² makes it quadratic |
2/x + 1 = 5 | ❌ No | x is in a denominator |
√x = 4 | ❌ No | Square root of variable |
|x − 3| = 7 | ⚠️ Special | Splits into two linear cases |
2. The 4-Step Solving Method
Use the following method to organize the algebra. After simplifying, check whether the equation has one solution, no solution, or infinitely many solutions.
STEP 1: CLEAR fractions or parentheses
STEP 2: COLLECT like terms on each side
STEP 3: MOVE variables to one side, numbers to the other
STEP 4: DIVIDE to isolate the variable
CHECK by plugging back in (optional, fast)
💡 Pro Tip: The 'Golden Rule' of Equations
Whatever you do to one side of the equation, you must do to the other side. Think of the equal sign as a balance scale: as long as both pans receive the same treatment, the scale stays level.
Operations you can safely apply to both sides: add, subtract, multiply by a non-zero number, divide by a non-zero number.
3. Worked Example: Method in Action
📝 Example: Solve 4(x − 3) + 5 = 2x + 11
Step 1: Distribute the 4: 4x − 12 + 5 = 2x + 11
Step 2: Combine like terms: 4x − 7 = 2x + 11
Step 3: Move variables left, numbers right: 4x − 2x = 11 + 7 → 2x = 18
Step 4: Divide both sides by 2: x = 9
Check: 4(9 − 3) + 5 = 24 + 5 = 29 and 2(9) + 11 = 29 ✓
4. Special Cases: No Solution vs. Infinite Solutions
On the SAT, some questions look ordinary but actually have either no solution or infinitely many. Compare the variable coefficients and constants after simplifying.
❓ After simplifying, do BOTH sides match exactly? | ||
✅ YES (e.g., 5 = 5) ▼ | ❌ NO (e.g., 5 = 7) ▼ | |
Infinite Solutions: every value of x works (an IDENTITY) | No Solution: the equation contradicts itself | |
⚡ Strategy: The Coefficient Test
When an SAT problem says "the equation has no solution" or "is true for all values of x", DO NOT solve it the long way.
Set up: Make the coefficients of x on both sides equal.
- If the constants then differ → NO SOLUTION
- If the constants are also equal → INFINITE SOLUTIONS (identity)
Example: ax + 7 = 5x + 7 has infinite solutions when a = 5. If a = 5 and the constants differed, it would have no solution.
5. Solving for One Variable in a Formula
To rearrange a formula, isolate the requested variable while treating the other quantities as constants. State any nonzero conditions before dividing.
📝 Example: Solve for h in V = π·r²·h
h is multiplied by π·r². To isolate h, divide both sides by π·r²:
h = V / (π · r²)
6. Five Useful Strategies
⚡ STRATEGY 1: Clear all fractions FIRST
Multiply every term by the least common denominator. This produces an equivalent equation without fractions when the denominators are nonzero constants.
⚡ STRATEGY 2: Plug in answer choices (back-solve)
With numerical answer choices, substitution can be useful. Test each candidate in the original equation and keep only values that satisfy it.
⚡ STRATEGY 3: Match coefficients for 'no solution / identity'
Compare the coefficient of x on each side, then compare the constants to classify the solutions.
⚡ STRATEGY 4: Skip distribution when both sides share a factor
If both sides have the same factor multiplied out, divide it away first. Example: 3(x+2) = 3(2x−1) → divide by 3 immediately.
⚡ STRATEGY 5: When asked for an EXPRESSION, don't solve for x
If the question asks for the value of (4x + 5), find what to multiply/add to your equation to MAKE that expression appear. Don't waste time solving for x first.
7. Common Mistakes to Avoid
⚠️ Common Mistakes
- Distributing a negative sign to only the first term inside parentheses.
- Dividing only one term on a side instead of every term.
- Subtracting a term from only one side. Apply the same subtraction to both sides.
- Multiplying by a denominator but skipping a term that doesn't have a fraction.
- Treating x/2 the same as 2x during cross-multiplication.
8. Summary
📌 Key Takeaways
- Definition: A linear equation has variables to the 1st power only.
- Method: Clear → Combine → Move → Divide → Check.
- Golden Rule: Do the same thing to both sides.
- No solution: Coefficients of x match but constants differ.
- Infinite solutions: Coefficients of x AND constants both match.
- Speed tip: Back-solve from answer choices when possible.
- Expression questions: Manipulate the equation directly: don't solve for x first.
9. Practice: 10 Practice Questions
⏱️ Instructions
Solve each question and show your reasoning. Check your work against the solutions at the end of this section.
Q1. If 3(x − 4) = 2x + 5, what is the value of x?
A) 7 B) 11 C) 17 D) 23
Hint: Distribute, then move x's to the left and numbers to the right in one quick step.
Q2. If (x + 5)/3 = 4, what is x?
A) 2 B) 7 C) 12 D) 17
Hint: Clear the denominator first by multiplying both sides by 3, then subtract.
Q3. For what value of k does the equation k·x + 8 = 4x + 8 have infinitely many solutions?
A) 0 B) 2 C) 4 D) 8
Hint: Use the Coefficient Test: both sides must match completely.
Q4. For what value of k does 5x + 7 = k·x + 9 have NO solution?
A) 5 B) 7 C) 9 D) any value
Hint: Coefficients of x must match while constants differ.
Q5. If 2x + 3 = 17, what is the value of 4x + 6?
A) 17 B) 28 C) 34 D) 40
Hint: Don't solve for x! Multiply the entire equation by 2.
Q6. If 1/2(x + 6) = 1/3(x + 12), what is x?
A) 0 B) 6 C) 12 D) 18
Hint: Multiply both sides by 6 to wipe out both fractions in one move.
Q7. If 7x − 3 = 5x + 11, what is the value of x − 7?
A) 0 B) 7 C) 14 D) −7
Hint: Solve for x quickly (x = 7), then subtract.
Q8. Solve for r in the formula C = 2π·r.
A) r = C·2π B) r = C/π C) r = C/(2π) D) r = 2π/C
Hint: Just divide both sides by 2π. No strategys: pure isolation.
Q9. If 3(2x + 4) − 5 = 2(3x + 7), the equation has…
A) one solution (x = 0) B) no solution C) infinite solutions D) x = 5
Hint: Expand both sides and compare. If you get something like 7 = 14, it's no solution.
Q10. If 4x + 7 = 19, what is the value of 12x + 21?
A) 36 B) 38 C) 57 D) 60
Hint: Multiply the whole equation by 3: you get 12x + 21.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | C) 17 | 3x − 12 = 2x + 5 → x = 17. |
Q2. | B) 7 | Multiply by 3: x + 5 = 12 → x = 7. |
Q3. | C) 4 | Coefficients must match: k = 4 (constants already match). |
Q4. | A) 5 | Coefficient match (k = 5), constants differ (7 ≠ 9) → no solution. |
Q5. | C) 34 | Multiply both sides of the original by 2: 4x + 6 = 34. |
Q6. | B) 6 | Multiply by 6: 3(x + 6) = 2(x + 12) → 3x + 18 = 2x + 24 → x = 6. |
Q7. | A) 0 | x = 7, so x − 7 = 0. |
Q8. | C) r = C/(2π) | Divide both sides by 2π. |
Q9. | B) no solution | LHS: 6x + 7. RHS: 6x + 14. 7 ≠ 14 → no solution. |
Q10. | C) 57 | Multiply original by 3: 12x + 21 = 57. |