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📝 Linear equations in two variables (6 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 6 questions available

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What is Linear equations in two variables?

Definition:
Linear equations in two variables are equations that can be written in the form $Ax + By = C$ , where $A, B, C$ are constants and $x, y$ are variables. They graph as straight lines on a coordinate plane and have infinitely many ordered pair solutions $(x, y)$ that satisfy them.

Working:
To find solutions, choose a value for one variable and solve for the other. For example, for $2x + y = 6$ , if $x = 1$ , then $2(1) + y = 6$ , so $y = 4$ , giving solution (1,4).

Example:
For $x - 3y = 9$ , if $y = 0$ , then $x = 9$ , solution (9,0). If $x = 0$ , then $-3y = 9$ , so $y = -3$ , solution (0, -3).

Reason:
Understanding two-variable equations is key to graphing, systems of equations, and modeling real-world relationships with two unknowns.

Easy

Medium

Hard

📝 All Linear equations in two variables MCQs

Q1. A student solves $6x-3y=21$ as follows: $6x-21=3y$ , then $y=2x-21$ . What is the specific error?

A.The student should add 21, not subtract it

B.The student divided only

6x

by 3 and failed to divide 21 by 3 ✅

C.The student should divide by 6 first

D.The student incorrectly changed

3y

into

-3y

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: After rearranging, $3y=6x-21$ . Dividing the entire right side by 3 gives $y=2x-7$ . The student's error is failing to divide the constant 21 by 3, a common misconception when isolating a variable.

Q2. Two students solve $8x+4y=32$ . Student A gets $y=8-2x$ , while Student B gets $y=2x-8$ . Which conclusion is correct?

A.Student A is correct because the

x

-term is subtracted before division ✅

B.Student B is correct because

4y

is positive

C.Both are correct for different values of

x

D.Neither is correct because

y

cannot be isolated

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Subtracting $8x$ gives $4y=32-8x$ . Dividing by 4 produces $y=8-2x$ . Student B reversed the signs incorrectly. Substituting a simple value such as $x=0$ confirms that $y=8$ , supporting Student A.

Q3. The graph of a linear equation crosses the $y$ -axis at 6 and passes through the point $(3,0)$ . Which equation represents the line in solved-for- $y$ form?

y=2x+6

y=-2x+6

✅

y=-\frac{1}{2}x+6

y=\frac{1}{2}x-6

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The $y$ -intercept is 6. Using the points $(0,6)$ and $(3,0)$ , the slope is $(0-6)/(3-0)=-2$ . Therefore the equation is $y=-2x+6$ , matching both graph features.

Q4. A company models its remaining inventory with $4x+y=120$ , where $x$ is the number of boxes sold and $y$ is inventory remaining. Which interpretation follows from solving for $y$ ?

y=120+4x

, so inventory rises as boxes are sold

y=120-4x

, so each additional box sold reduces inventory by 4 units ✅

y=4x-120

, so inventory starts negative

y=\frac{120}{4}-x

, so each box reduces inventory by 1

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Rearranging $4x+y=120$ gives $y=120-4x$ . The intercept 120 represents the starting inventory, while the coefficient $-4$ means every additional box sold reduces the remaining inventory by four units.

Q5. For $3x-2y=12$ , a learner substitutes $x=2y$ and obtains $y=6$ . Is this conclusion valid?

A.Yes, because

3(2y)-2y=12

gives

y=6

B.No, because substituting

x=2y

gives

4y=12

, so

y=3

✅

C.No, because

x=2y

cannot be substituted into a linear equation

D.Yes, because

x

and

y

must have equal coefficients

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Substituting $x=2y$ gives $3(2y)-2y=12$ , so $6y-2y=12$ , which simplifies to $4y=12$ and therefore $y=3$ . The learner's arithmetic conclusion $y=6$ is incorrect.

Q6. A line has equation $9x+3y=27$ . Another student rewrites it as $y=9-3x$ . Which equivalent form correctly preserves the line and best reveals its slope?

y=9-3x

✅

y=3-3x

y=9x-9

y=3x-9

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Subtracting $9x$ from both sides gives $3y=27-9x$ . Dividing every term by 3 produces $y=9-3x$ . Therefore A is mathematically correct, while B incorrectly divides the constant term. The slope is $-3$ .

🔗 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:Solving for a variable in two variable equation📝 Linear equations in two variablesHow to Solve Linear Inequalities:Next Topic→