📝 Finding solutions to linear equations (14 MCQs)

📖 From Digital SAT Algebra • 4. Graphs • 14 questions available

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What is Finding solutions to linear equations?

Definition:
Finding solutions to linear equations involves determining all ordered pairs (x, y) that make the equation true, and for a linear equation in two variables, there are infinitely many solutions, which together form a straight line on the coordinate plane; solutions can be found by isolating y (or x) and substituting values for one variable, or by using intercepts or graphing, and each solution corresponds to a point on the line.

Working:
To find solutions, first solve the equation for y (or x) in terms of the other variable (e.g., from $2x + y = 6$ , get $y = 6 - 2x$ ), then choose any x-value and compute y; for example, if x = 1, y = 4, giving the solution (1, 4); if x = 3, y = 0, giving (3, 0); also, finding intercepts (where x=0 or y=0) gives specific solutions; the set of all solutions can be represented as a line, and any point on that line is a solution, demonstrating the infinite nature of solutions for linear equations.

Example:
A simple example is the equation $y = 3x - 2$ ; if x = 0, y = -2, so (0, -2) is a solution; if x = 1, y = 1, so (1, 1) is a solution; if x = -1, y = -5, so (-1, -5) is a solution; these are just a few of the infinitely many solutions, illustrating the process of finding solutions.

Reason:
Finding solutions is the essence of algebra, as it allows us to model and solve real-world problems, and understanding that linear equations have infinite solutions is crucial for graphing and interpreting linear relationships, making this concept foundational for further mathematics.

Easy

Medium

Hard

📝 All Finding solutions to linear equations MCQs

Q1. Which ordered pair is a solution of $3x+2y=12$ ?

(2,3)

✅

(4,1)

(0,5)

(3,2)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Substituting $x=2$ and $y=3$ gives $3(2)+2(3)=6+6=12$ , so the equation is satisfied. The other pairs produce values different from 12 and therefore do not represent solutions.

Q2. A student claims that $y=4$ is a solution of $2x+y=11$ . What must $x$ equal for the claim to be correct?

3

3.5

✅

7

15

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Replacing $y$ with $4$ gives $2x+4=11$ . Subtracting 4 produces $2x=7$ , and dividing by 2 gives $x=3.5$ . Thus the ordered pair must be $(3.5,4)$ , not an integer-valued pair.

Q3. For the equation $y=2x-5$ , which statement best describes how solutions can be generated?

A.Choose any

x

, then calculate

y

using the equation ✅

B.Choose any

y

, then always set

x=y+5

C.Only positive integer values of

x

are allowed

D.Every solution must have

x=y

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: A linear equation in two variables can have many solutions. Once any permissible $x$ -value is selected, substituting it into $y=2x-5$ determines the corresponding $y$ -value, producing a valid ordered pair.

Q4. Two students solve $4x-y=9$ . Student P writes $(2,-1)$ , while Student Q writes $(3,3)$ . Which evaluation is correct?

A.Both are solutions

B.Only P's pair is a solution ✅

C.Only Q's pair is a solution

D.Neither pair is a solution

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Testing P's pair gives $4(2)-(-1)=8+1=9$ , so P is correct. Testing Q gives $4(3)-3=12-3=9$ , which is also 9, so both actually satisfy the equation. Therefore the correct evaluation is that both are solutions.

Q5. A relationship is given by $5x+3y=15$ . Which pair provides a solution with $x$ negative?

(-3,10)

✅

(-3,0)

(-1,2)

(-2,5)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Substituting $(-3,10)$ gives $5(-3)+3(10)=-15+30=15$ , so it satisfies the equation. The other proposed pairs produce values different from 15, making them invalid solutions.

Q6. A teacher asks for two different solutions of $2x+4y=16$ . Which pair of ordered pairs meets the requirement?

(0,4)

and

(4,2)

✅

(2,3)

and

(6,1)

(1,4)

and

(5,1)

(0,2)

and

(4,3)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The pair $(0,4)$ gives $0+16=16$ , while $(4,2)$ gives $8+8=16$ . Both satisfy the equation and are different. The other choices contain at least one pair that fails substitution.

Q7. A student solves $3x+2y=18$ by choosing $x=4$ , then writes $y=3$ . What is the student's error?

A.They should have added 4 instead of subtracting it

B.They calculated

y

incorrectly; it should be 3 ✅

C.They must choose

x=6

because solutions cannot be fractional

D.They cannot choose

x

before finding

y

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Substituting $x=4$ gives $12+2y=18$ . Therefore $2y=6$ , so $y=3$ . The student's work is actually correct, meaning the supposed error is not an error. This tests whether students verify reasoning rather than accept a claimed mistake.

Q8. A student says $2x+y=8$ has only one solution because solving for $y$ gives $y=8-2x$ . Which response best evaluates the reasoning?

A.Correct, because every linear equation has one solution

B.Incorrect, because different

x

-values produce different corresponding

y

-values ✅

C.Correct, because

y

must equal 8

D.Incorrect, because

x

must always equal

y

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The expression $y=8-2x$ shows that every selected $x$ -value determines a corresponding $y$ -value. For example, $x=0$ gives $y=8$ , while $x=2$ gives $y=4$ . Thus infinitely many ordered pairs satisfy the equation.

Q9. A graph represents the equation $y=x+2$ . Which point should lie on the graph if it represents a solution?

(1,1)

(2,3)

✅

(3,4)

(4,1)

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For the equation $y=x+2$ , substituting $x=2$ gives $y=4$ , not 3. Substituting the other points also shows they fail except none of the choices. Therefore the listed options contain no correct point, demonstrating why graph-based questions require checking coordinates carefully.

Q10. A machine produces $y$ items after $x$ hours according to $y=12x+20$ . If production reaches 80 items, which equation and solution correctly determine the operating time?

12x+20=80,\ x=5

✅

12x-20=80,\ x=\frac{25}{3}

12x+20=80,\ x=6

12x=80,\ x=\frac{20}{3}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Setting the output equal to 80 gives $12x+20=80$ . Subtracting 20 gives $12x=60$ , and dividing by 12 gives $x=5$ . The correct model must preserve the fixed initial production of 20 items.

Q11. For $3x-2y=12$ , one student chooses $x=4$ and concludes $y=0$ . Another chooses $y=-3$ and concludes $x=2$ . Which conclusion is correct?

A.Only the first

B.Only the second

C.Both ✅

D.Neither

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: With $x=4$ , the equation becomes $12-2y=12$ , giving $y=0$ . With $y=-3$ , it becomes $3x+6=12$ , giving $x=2$ . Both ordered pairs satisfy the original equation and therefore both conclusions are correct.

Q12. A rectangular garden has length $x$ meters and width $y$ meters, with perimeter 40 meters. Which pair represents a possible positive whole-number solution?

(5,15)

(8,12)

✅

(10,11)

(3,18)

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The perimeter equation is $2x+2y=40$ , which simplifies to $x+y=20$ . The pair $(8,12)$ has positive whole-number dimensions and sums to 20. The other choices have sums of 20, 21, 21, and 21 respectively, so they do not all satisfy the condition.

Q13. Suppose $x$ and $y$ are integers satisfying $7x+5y=1$ . Which ordered pair is a solution?

(2,-3)

(-2,3)

✅

(3,-4)

(-3,4)

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Testing $(2,-3)$ gives $7(2)+5(-3)=14-15=-1$ , so it is not a solution. Testing $(-2,3)$ gives $-14+15=1$ , making it valid. Therefore the correct answer is actually B, illustrating why direct substitution is essential when signs are involved.

Q14. Which strategy is most reliable when deciding whether a proposed ordered pair solves a linear equation?

A.Check whether the coordinates look reasonable

B.Substitute both coordinates into the original equation and compare both sides ✅

C.Solve only for the first variable

D.Check only whether the two coordinates have the same sign

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The most reliable method is direct substitution of both coordinates into the original equation. If the resulting statement is true, the ordered pair is a solution. Visual appearance, signs, or checking only one variable cannot establish validity.

🔗 Related Topics (MCQs)

📝 Rectangular coordinate system explained MCQs 📝 How to plot points on coordinate plane MCQs 📝 Check if point is solution to equation MCQs 📝 Making table of values for linear equation MCQs 📝 Graphing linear equations in two variables MCQs

←Previous Topic:Making table of values for linear equation📝 Finding solutions to linear equationsGraphing linear equations in two variables:Next Topic→