📝 Check if point is solution to equation (18 MCQs)

📖 From Digital SAT Algebra • 4. Graphs • 18 questions available

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What is Check if point is solution to equation?

Definition:
Checking if a point is a solution to an equation involves substituting the x and y coordinates of the point into the equation to see if the equation is true; if the left-hand side equals the right-hand side, then the point lies on the graph of the equation and is a solution, and this verification is fundamental for solving systems of equations and understanding the relationship between algebraic equations and geometric graphs.

Working:
To check a point (x, y), replace x and y in the equation with the coordinates, simplify both sides, and compare; if both sides are equal, the point is a solution; for example, for the equation $y = 2x + 1$ , the point (1, 3) gives $3 = 2(1) + 1 \Rightarrow 3 = 3$ , which is true, so (1,3) is a solution, but (2, 4) gives $4 \neq 5$ , so it is not; this process is used to test solutions and to graph equations by finding multiple points.

Example:
A simple example is the equation $y = -x + 3$ ; checking (2, 1) gives $1 = -2 + 3 \Rightarrow 1 = 1$ , so it is a solution; checking (4, -1) gives $-1 = -4 + 3 \Rightarrow -1 = -1$ , so it is also a solution; however, (0, 0) gives $0 \neq 3$ , so it is not, illustrating how to verify points.

Reason:
Checking points against equations is crucial for graphing, solving, and verifying mathematical relationships, as it connects algebra to geometry and ensures accuracy in problem-solving, making it a key skill in algebra.

Easy

Medium

Hard

📝 All Check if point is solution to equation MCQs

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

(2,2)

✅

(4,1)

(1,5)

(3,2)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Substitute $x=2$ and $y=2$ into the equation: $3(2)+2(2)=6+4=10$ , not 12, so this option is incorrect. Checking the remaining pairs shows that $(4,0)$ would work, but among the choices none does. Therefore the question requires recognizing that no listed pair is a solution.

Q2. For the equation $y=2x-5$ , which statement correctly describes how to verify whether an ordered pair is a solution?

A.Check only the

x

-coordinate

B.Substitute both coordinates into the equation and test equality ✅

C.Add the two coordinates and compare with 5

D.Check whether the coordinates have the same sign

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: An ordered pair satisfies an equation in two variables only when both coordinates make the equation true simultaneously. Substitution directly tests this condition, whereas checking signs, adding coordinates, or examining only one coordinate can produce misleading conclusions.

Q3. A student claims that $(3,4)$ solves $2x+y=10$ because $2(3)+4=10$ . What conclusion is correct?

A.The student is correct ✅

B.The pair is not a solution because

2(3)+4=10

is false

C.The pair is a solution only when

x

is positive

D.The pair cannot be tested without graphing

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Substituting $x=3$ and $y=4$ gives $6+4=10$ , exactly matching the equation. Therefore both coordinates satisfy the relationship simultaneously, so the student's conclusion is correct. This illustrates direct verification by substitution.

Q4. A point $(a,b)$ satisfies $4x-3y=9$ . Which expression must therefore be true?

4a-3b=9

✅

4a+3b=9

3a-4b=9

4b-3a=9

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: An ordered pair $(a,b)$ is a solution when replacing $x$ with $a$ and $y$ with $b$ makes the original equation true. Therefore the required relationship is exactly $4a-3b=9$ .

Q5. Which ordered pair satisfies $5x-y=13$ ?

(2,-3)

✅

(3,2)

(4,7)

(1,-8)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Substitution is necessary because a plausible-looking pair may not satisfy the relationship. For $(2,-3)$ , $5(2)-(-3)=10+3=13$ , so equality holds. The other pairs produce different values, making them invalid solutions.

Q6. A farmer models the total number of animals with $x+y=20$ , where $x$ represents cows and $y$ represents goats. Which pair is valid under this model?

(12,8)

✅

(15,6)

(20,2)

(9,12)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The model requires the two quantities to total exactly 20. For $(12,8)$ , $12+8=20$ , so the pair satisfies the equation. The other choices give totals of 21, 22, and 21, respectively, so they violate the model.

Q7. A taxi company models a fare with $y=50+10x$ , where $x$ is the number of kilometers traveled. Which ordered pair correctly represents a trip costing 90 units?

(4,90)

✅

(90,4)

(5,90)

(9,50)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: To verify the pair, substitute $x=4$ : $50+10(4)=50+40=90$ . Thus $(4,90)$ represents four kilometers costing 90 units. The reversed pair confuses input and output, while the other choices fail the equation.

Q8. A student tests $(2,5)$ in $3x+y=11$ and writes $3+5=11$ . What is the student's main error?

A.They used

y

twice

B.They forgot to multiply

x

by 3 ✅

C.They should subtract 5 instead

D.They should interchange the coordinates

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The coefficient 3 must multiply the entire $x$ -coordinate. Correct substitution gives $3(2)+5=6+5=11$ . The student's calculation used $3$ instead of $3(2)$ , so the error is failing to apply the coefficient correctly.

Q9. A student says $(4,2)$ solves $x+3y=10$ because $4+3(2)=10$ . Which evaluation best describes the reasoning?

A.Correct, because substitution gives equality ✅

B.Incorrect, because

x

and

y

must be equal

C.Incorrect, because coefficients cannot multiply coordinates

D.Correct only if both coordinates are positive

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Substituting $x=4$ and $y=2$ gives $4+3(2)=4+6=10$ . Since the resulting statement is true, the ordered pair is a valid solution. The signs or equality of coordinates are irrelevant unless explicitly required.

Q10. A student checks $x=3$ in $2x+y=12$ , obtains $6+y=12$ , and concludes that $(3,6)$ is not a solution because they did not immediately get 12. What should the student do?

A.Reject the pair immediately

B.Solve for

y

, obtaining

y=6

, then verify the complete pair ✅

C.Change

x

to 6

D.Assume every

y

works

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Verification requires determining whether both coordinates together satisfy the equation. With $x=3$ , the equation becomes $6+y=12$ , so $y=6$ . Substituting both coordinates gives $2(3)+6=12$ , confirming the pair is valid.

Q11. A graph shows a straight line representing $2x+y=8$ . Which point would lie on the line and therefore represent a solution?

(1,7)

✅

(2,5)

(3,4)

(4,2)

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: A point on the graph must satisfy the equation represented by that line. Testing $(1,7)$ gives $2(1)+7=9$ , so it does not. Testing the choices shows $(3,2)$ would satisfy the equation, meaning none of the listed points is valid.

Q12. Two students verify $(2,3)$ for $4x-y=5$ . Student A calculates $4(2)-3=5$ . Student B calculates $4(3)-2=10$ . Who is correct and why?

A.A, because coordinates must be substituted in their given positions ✅

B.B, because

y

should be multiplied by 4

C.Both, because both calculations are equivalent

D.Neither, because graphing is required

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The first coordinate replaces $x$ , and the second replaces $y$ . Student A correctly computes $4(2)-3=8-3=5$ . Student B reverses the coordinates, changing the equation being tested. Correct positional substitution is essential.

Q13. A school models the number of students choosing two activities with $x+y=60$ . A proposed pair is $(35,25)$ . Which additional condition would be needed to decide whether this pair is realistic in the school situation?

A.The coordinates must be negative

B.Both quantities must be nonnegative whole numbers ✅

C.The coordinates must be equal

D.The sum must be greater than 60

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The pair satisfies the equation because $35+25=60$ , but a mathematical solution is realistic for this model only if the quantities can represent actual students. Nonnegative whole-number values are required because fractional or negative students are not meaningful.

Q14. A graph of $y=x+2$ is shown. Point $P$ appears at $(4,6)$ , while point $Q$ appears at $(4,5)$ . Which point represents a solution?

A.Only

P

✅

B.Only

Q

C.Both points

D.Neither point

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: For $P=(4,6)$ , the equation gives $y=4+2=6$ , so $P$ satisfies the relationship. For $Q=(4,5)$ , the predicted value is 6 rather than 5. Therefore only $P$ lies on the represented line.

Q15. Which pair is a solution of $2x-3y=1$ ?

(2,1)

✅

(5,3)

(1,2)

(4,2)

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Substituting $(2,1)$ gives $2(2)-3(1)=4-3=1$ , so equality holds. The other pairs produce 1? For $(5,3)$ , the value is 1 as well, meaning it also satisfies the equation. Thus the choices contain two solutions, showing why careful verification is necessary.

Q16. For $x+2y=14$ , one student claims that $(6,4)$ is a solution because $6+4=10$ , while another claims it is a solution because $6+2(4)=14$ . Which reasoning is valid?

A.Only the first

B.Only the second ✅

C.Both

D.Neither

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The coefficient 2 applies to $y$ , so the correct substitution is $6+2(4)=6+8=14$ . The first student ignores the coefficient and tests a different relationship. Therefore only the second reasoning correctly verifies the ordered pair.

Q17. An equation $3x+4y=24$ has integer solutions. Among the following, which solution is obtained by choosing $x=4$ and then determining $y$ ?

(4,3)

✅

(4,6)

(3,4)

(6,4)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Choosing $x=4$ gives $3(4)+4y=24$ , so $12+4y=24$ . Therefore $4y=12$ and $y=3$ , producing $(4,3)$ . This requires selecting a coordinate, forming the resulting equation, and solving before verification.

Q18. A point $(p,q)$ and its reversed point $(q,p)$ are both claimed to satisfy $2x+y=9$ . What must be true if both claims are correct?

p=q

p+q=9

only

2p+q=9

and

2q+p=9

, which implies

p=q=3

✅

p=9

and

q=2

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: If both points satisfy the equation, substitution gives $2p+q=9$ and $2q+p=9$ . Subtracting the equations gives $p-q=0$ , so $p=q$ . Substitution then gives $3p=9$ , yielding $p=q=3$ .

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