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📝 Make the coefficient of the variable equal to 1 (11 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 11 questions available

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What is Make the coefficient of the variable equal to 1?

Definition:
Making the coefficient of the variable equal to 1 means transforming an equation of the form $ax = b$ into $x = b/a$ by dividing both sides by the coefficient $a$ (or multiplying by its reciprocal). This final step isolates the variable completely, giving the solution. This is achieved using the Division or Multiplication Property of Equality.

Working:
Given $7x = 42$ , divide both sides by 7 to get $x = 6$ . If the coefficient is a fraction, like $\frac{2}{3}x = 8$ , multiply both sides by the reciprocal $\frac{3}{2}$ : $x = 8 \cdot \frac{3}{2} = 12$ .

Example:
Solve $5y = 45$ . Divide both sides by 5: $5y/5 = 45/5$ , so $y = 9$ . Verify: $5(9) = 45$ .

Reason:
This step is essential because the solution of an equation is the value of the variable alone, so making the coefficient 1 gives the direct answer in its simplest form.

Easy

Medium

Hard

📝 All Make the coefficient of the variable equal to 1 MCQs

Q1. A student solves the equation $7x = 42$ by dividing both sides by 7, getting $x = 6$ . Which property of equality justifies this step, and what is the primary purpose of this action?

A.The Addition Property of Equality; to isolate the variable

B.The Multiplication Property of Equality; to make the coefficient of

x

equal to 1 ✅

C.The Distributive Property; to remove the parentheses

D.The Subtraction Property of Equality; to combine like terms

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The action of dividing both sides by 7 is an application of the Multiplication Property of Equality (specifically, multiplying by $1/7$ ). The primary goal is to transform the coefficient of the variable from 7 to 1, thereby isolating the variable to find its value.

Q2. In solving the equation $-4y = 20$ , a student writes the next step as $y = 20 - 4$ . What is the error in this reasoning, and what should the correct step be?

A.The student incorrectly applied the Subtraction Property; they should have divided both sides by -4 to get

y = -5

. ✅

B.The student correctly isolated the variable; their solution

y = 16

is correct.

C.The student should have added 4 to both sides to get

y = 24

D.The student should have multiplied both sides by -4 to get

y = -80

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The student's step $y = 20 - 4$ treats the coefficient $-4$ as a term to be subtracted, not as a multiplier. The correct operation is division: $-4y / -4 = 20 / -4$ , which simplifies to $y = -5$ . This common mistake shows a lack of understanding of inverse operations.

Q3. A car rental company charges a flat fee of $30 plus$ 0.15 per mile driven. If the total bill is $C$ dollars and the number of miles driven is $m$ , the equation is $C = 30 + 0.15m$ . If a customer's bill is $75, which equation represents the first step to find $m$ by making the coefficient of $m$ equal to 1?

0.15m = 75

m = \frac{75 - 30}{0.15}

0.15m = 75 - 30

✅

m = \frac{75}{0.15} - 30

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: To solve $75 = 30 + 0.15m$ , the first step is to isolate the term with $m$ by subtracting 30 from both sides, resulting in $45 = 0.15m$ or $0.15m = 45$ . The coefficient of $m$ (0.15) is then made 1 by dividing both sides by 0.15. Option C correctly shows this first step.

Q4. Two students solve the equation $\frac{2}{3}x = 8$ . Student A multiplies both sides by $\frac{3}{2}$ . Student B divides both sides by $\frac{2}{3}$ . Which statement is true about their methods and results?

A.Both students will get the same correct answer,

x = 12

, as both operations are valid. ✅

B.Student A is correct, Student B is incorrect because you cannot divide by a fraction.

C.Student A will get

x = 12

, and Student B will get a different, incorrect answer.

D.Both students are incorrect; they should multiply both sides by 2 and then divide by 3.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: To make the coefficient of $x$ equal to 1, you can either multiply by the reciprocal of $\frac{2}{3}$ , which is $\frac{3}{2}$ , or divide by $\frac{2}{3}$ . Since dividing by a fraction is the same as multiplying by its reciprocal, both operations are mathematically equivalent and will correctly yield $x = 8 \times \frac{3}{2} = 12$ .

Q5. The graph of the line $y = 3x + 2$ is shown. A student wants to find the x-intercept. They set $y=0$ and get $0 = 3x + 2$ . What is the next step to solve for $x$ and what is the x-intercept?

A.Subtract 2 from both sides, then divide by 3 to get

x = -\frac{2}{3}

. ✅

B.Add 2 to both sides, then multiply by 3 to get

x = 6

C.Divide both sides by 3, then subtract 2 to get

x = -\frac{2}{3}

D.Subtract 2 from both sides, then multiply by 3 to get

x = -6

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The x-intercept occurs where the line crosses the x-axis ( $y=0$ ). To solve $0 = 3x + 2$ , the first step is to subtract 2 from both sides to get $-2 = 3x$ . Then, to make the coefficient of $x$ equal to 1, divide both sides by 3, resulting in $x = -2/3$ . This point $(-2/3, 0)$ is the x-intercept.

Q6. In the equation $ax + b = c$ , where $a$ , $b$ , and $c$ are constants and $a \neq 0$ , what is the result of making the coefficient of $x$ equal to 1, and what does this final form represent?

x = \frac{c-b}{a}

, which is the solution for

x

. ✅

x = c - b - a

, which is the solution for

x

x = \frac{c}{a} - b

, which is the solution for

x

x = \frac{c-b}{a}

, which is the equation of a line.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: First, subtract $b$ from both sides to isolate the term with $x$ : $ax = c - b$ . Then, divide both sides by $a$ (which makes the coefficient of $x$ equal to 1) to get $x = (c-b)/a$ . This final form explicitly gives the value of $x$ that satisfies the original equation.

Q7. A student is trying to solve $2(x+3) = 14$ . They have two options: Option A: Divide both sides by 2 first, getting $x+3 = 7$ . Option B: Distribute the 2 first, getting $2x + 6 = 14$ , then subtract 6 and divide by 2. Which approach is more efficient and why?

A.Option A is more efficient because it makes the coefficient of the inner expression equal to 1 first, simplifying the equation.

B.Option B is more efficient because it correctly applies the distributive property.

C.Both are equally efficient and mathematically valid, but Option A avoids extra steps. ✅

D.Option A is incorrect because you cannot divide the entire equation by 2 when there is addition inside parentheses.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Both methods are mathematically sound. Option A uses the division property of equality to make the coefficient of the binomial $x+3$ equal to 1, simplifying to $x+3=7$ . Option B uses distribution. However, Option A is generally more efficient as it reduces the number of steps. This demonstrates flexibility in approaching linear equations.

Q8. Solve for $x$ : $0.25x = 10$ . A student suggests multiplying both sides by 100 to eliminate the decimal, then dividing by 25. Is this a valid method to make the coefficient of $x$ equal to 1, and what is the solution?

A.Yes, it's valid;

0.25x = 10

becomes

25x = 1000

, then

x = 40

. ✅

B.No, it's invalid because you must multiply both sides by 4, not 100.

C.Yes, it's valid;

0.25x = 10

becomes

25x = 10

, then

x = 0.4

D.No, it's invalid because you can only multiply by the reciprocal of 0.25.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Multiplying both sides by 100 to clear the decimal is a valid first step, resulting in $25x = 1000$ . Then, to make the coefficient of $x$ equal to 1, divide both sides by 25, giving $x = 40$ . This method achieves the same result as multiplying by 4 (the reciprocal of 0.25), demonstrating that multiple strategies can work.

Q9. Which of the following equations is NOT in a form where the coefficient of the variable is 1, but can be solved by first making it 1 using a single operation?

-x = 5

x + 3 = 7

✅

\frac{x}{4} = 2

2x = 8

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: In option B, the coefficient of $x$ is already 1. You do not need to make it 1; you simply subtract 3 from both sides to solve for $x$ . The other options have coefficients of -1, 1/4, and 2, which need to be made equal to 1 by multiplying or dividing by a constant.

Q10. Find the value of $x$ in the equation $\frac{3x - 5}{2} = 11$ . To solve, a student first multiplies both sides by 2, then adds 5, then divides by 3. What is the rationale behind the order of these steps, and what is the solution?

A.The order is arbitrary; any order will yield the correct answer.

B.The order is designed to reverse the order of operations; first undo division, then subtraction, then multiplication to make coefficient 1. ✅

C.The order must be: divide by 3 first, then add 5, then multiply by 2.

D.The order is to first make the coefficient of x equal to 1, then undo other operations.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Following the reverse order of operations (PEMDAS in reverse), you first undo the division by 2 (multiply by 2), then undo the subtraction of 5 (add 5), and finally undo the multiplication by 3 (divide by 3) to make the coefficient of $x$ equal to 1. This yields $3x - 5 = 22$ , $3x = 27$ , $x = 9$ .

Q11. A gardener has a rectangular plot with length $L$ meters and width $W$ meters. The perimeter is given by $P = 2L + 2W$ . If the perimeter is 100 meters and the width is 20 meters, the equation becomes $100 = 2L + 40$ . What is the first step to solve for $L$ by making the coefficient of $L$ equal to 1?

A.Subtract 40 from both sides, then divide by 2.

B.Divide both sides by 2, then subtract 20. ✅

C.Add 40 to both sides, then multiply by 2.

D.Subtract 2L from both sides, then divide by 40.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Given $100 = 2L + 40$ , you can first subtract 40 to get $60 = 2L$ , then divide by 2 to get $L = 30$ . However, an alternate efficient first step is to divide the entire equation by 2, making the coefficient of $L$ equal to 1 immediately: $50 = L + 20$ . Then, subtract 20 to get $L = 30$ . This shows a more strategic application of making the coefficient 1 first.

🔗 Related Topics (MCQs)

📝 Subtraction and Addition Properties of Equality MCQs 📝 How to verify a solution to an equation MCQs 📝 Subtraction Property of Equality MCQs 📝 Addition Property of Equality MCQs 📝 Solving equations with simplification MCQs

←Previous Topic:Collect constant terms on the other side📝 Make the coefficient of the variable equal to 1Classify equations as: Conditional equations (one or more solutions):Next Topic→