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📝 Solving for a variable in two variable equation (7 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 7 questions available

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What is Solving for a variable in two variable equation?

Definition:
Solving for a variable in a two-variable equation means expressing one variable explicitly in terms of the other. This is common in linear equations like $2x + 3y = 12$ . To solve for $y$ , isolate it: $3y = 12 - 2x$ , then $y = (12 - 2x)/3$ .

Working:
Given $4x - 5y = 20$ , to solve for $x$ , add $5y$ : $4x = 20 + 5y$ , divide by 4: $x = (20 + 5y)/4$ . To solve for $y$ , subtract $4x$ : $-5y = 20 - 4x$ , divide by -5: $y = (4x - 20)/5$ .

Example:
For $3a + 2b = 18$ , solve for $a$ : subtract $2b$ : $3a = 18 - 2b$ , divide by 3: $a = (18 - 2b)/3$ .

Reason:
This is foundational for graphing linear equations and substitution methods in systems of equations.

Easy

Medium

Hard

📝 All Solving for a variable in two variable equation MCQs

Q1. Which expression correctly solves $3x+2y=14$ for $y$ ?

y=7-3x

y=7-\frac{3}{2}x

y=\frac{14-3x}{2}

D.Both B and C are equivalent ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: To isolate $y$ , subtract $3x$ from both sides to obtain $2y=14-3x$ , then divide by 2. This gives $y=7-\frac{3}{2}x$ , which is algebraically identical to $\frac{14-3x}{2}$ .

Q2. For $5x-y=12$ , what is the most efficient expression for $y$ in terms of $x$ ?

y=5x-12

✅

y=12-5x

y=5x+12

y=\frac{12}{5}-x

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Adding $y$ to both sides and subtracting 12 gives $5x-12=y$ . Therefore $y=5x-12$ . The other choices result from reversing a sign, adding instead of subtracting, or incorrectly dividing by 5.

Q3. A student claims that $4x+6y=18$ becomes $y=3-4x$ . Which reasoning best evaluates the claim?

A.It is correct because

18\div6=3

B.It is incorrect because

4x

must first be subtracted before dividing by 6 ✅

C.It is incorrect because 18 should be divided by 4 first

D.It is correct because coefficients can be divided independently

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The coefficient of $x$ must remain attached to $x$ when it is moved. Subtracting $4x$ gives $6y=18-4x$ , and dividing every term by 6 produces $y=3-\frac{2}{3}x$ , not $3-4x$ .

Q4. Which form of $7x-4y=20$ makes the value of $y$ easiest to determine when $x$ is known?

y=\frac{7x-20}{4}

✅

y=\frac{20-7x}{4}

y=20-7x

y=\frac{20+7x}{4}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Moving $7x$ to the other side gives $-4y=20-7x$ . Dividing by $-4$ changes both signs, yielding $y=\frac{7x-20}{4}$ . Wait carefully: this shows A is actually equivalent to the correct form, so the correct answer is A.

Q5. A line is described by $2x+5y=25$ . If $x$ increases from 0 to 5, how does $y$ change?

y

increases by 2

y

decreases by 2 ✅

y

decreases by 5

y

increases by 5

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Solving gives $y=5-\frac{2}{5}x$ . When $x$ increases by 5, the term $\frac{2}{5}x$ increases by 2, so $y$ must decrease by 2. This connects algebraic isolation with the relationship between variables.

Q6. A theater charges $x$ dollars for an adult ticket and $y$ dollars for a student ticket. A group buys 3 adult and 4 student tickets for 54 dollars. Which equation correctly expresses the student price $y$ in terms of $x$ ?

y=54-3x

y=\frac{54-3x}{4}

✅

y=\frac{54+3x}{4}

y=\frac{3x-54}{4}

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The total-cost equation is $3x+4y=54$ . Subtracting $3x$ gives $4y=54-3x$ , and dividing by 4 gives $y=\frac{54-3x}{4}$ . This models the situation while preserving the meaning of each variable.

Q7. A rectangle has perimeter 46 units, with length $x$ and width $y$ . Which expression represents $y$ in terms of $x$ , and what width results when $x=15$ ?

y=46-2x; y=16

y=23-x; y=8

✅

y=23-2x; y=-7

y=\frac{46-x}{2}; y=15.5

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The perimeter equation is $2x+2y=46$ . Dividing by 2 gives $x+y=23$ , so $y=23-x$ . Substituting $x=15$ gives $y=8$ , which is a realistic positive width.

🔗 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

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