📝 How to find x and y intercepts from equation (13 MCQs)

📖 From Digital SAT Algebra • 4. Graphs • 13 questions available

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What is How to find x and y intercepts from equation?

Definition:
Finding x and y intercepts from an equation involves setting the opposite variable to zero and solving for the remaining variable: for the x-intercept, set y = 0 and solve for x; for the y-intercept, set x = 0 and solve for y; these intercepts are critical points on the graph and are particularly easy to find from equations in standard form $Ax + By = C$ , providing key points for graphing.

Working:
For the x-intercept, substitute y = 0 into the equation and solve for x: $Ax + B(0) = C \Rightarrow x = \frac{C}{A}$ ; for the y-intercept, substitute x = 0 and solve for y: $A(0) + By = C \Rightarrow y = \frac{C}{B}$ ; for example, in $3x + 4y = 12$ , x-intercept is $3x = 12 \Rightarrow x = 4$ , so (4, 0); y-intercept is $4y = 12 \Rightarrow y = 3$ , so (0, 3); these intercepts are then plotted to graph the line.

Example:
A simple example is $5x - 2y = 10$ ; x-intercept: set y = 0, $5x = 10 \Rightarrow x = 2$ , so (2, 0); y-intercept: set x = 0, $-2y = 10 \Rightarrow y = -5$ , so (0, -5); these intercepts are easily found and used to graph the equation.

Reason:
Finding intercepts from equations is a fundamental algebraic skill that makes graphing efficient and helps in understanding the behavior of linear equations, and it is widely used in many fields for quickly sketching lines and analyzing relationships.

Easy

Medium

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📝 All How to find x and y intercepts from equation MCQs

Q1. For the line $3x+2y=12$ , a student finds the x-intercept by setting $x=0$ . Which statement correctly identifies the mistake and the correct x-intercept?

A.The student should set

y=0

, giving

(4,0)

. ✅

B.The student should set

x=1

, giving

(3,1.5)

C.The student should set

y=1

, giving

(10/3,1)

D.There is no x-intercept because both variables occur.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: An x-intercept occurs where the graph crosses the x-axis, and every point on that axis has $y=0$ . Substituting $y=0$ into $3x+2y=12$ gives $3x=12$ , so $x=4$ and the intercept is $(4,0)$ .

Q2. For $5x-4y=20$ , which ordered pair gives the y-intercept?

(4,0)

(0,-5)

(0,5)

✅

(-5,0)

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The y-intercept is found by setting $x=0$ , because points on the y-axis have zero x-coordinate. Substitution gives $-4y=20$ , so $y=-5$ . Therefore, the correct ordered pair is $(0,-5)$ , not $(0,5)$ .

Q3. Two students analyze $2x+3y=18$ . Student A says the intercepts are $(9,0)$ and $(0,6)$ . Student B says they are $(6,0)$ and $(0,9)$ . Which evaluation is correct?

A.Student A is correct because the coefficients become the intercepts. ✅

B.Student B is correct because

x=6

and

y=9

satisfy the equation separately.

C.Both are correct because either pair can represent the intercepts.

D.Neither is correct because both variables must be nonzero.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: For the x-intercept, set $y=0$ : $2x=18$ , giving $x=9$ . For the y-intercept, set $x=0$ : $3y=18$ , giving $y=6$ . Thus Student A correctly applies the axis conditions, while Student B reverses the roles of the coefficients.

Q4. A line has equation $4x-2y=8$ . Without fully solving for $y$ , which pair of intercepts must be obtained?

A.x-intercept

(2,0)

, y-intercept

(0,-4)

B.x-intercept

(-2,0)

, y-intercept

(0,4)

C.x-intercept

(2,0)

, y-intercept

(0,4)

✅

D.x-intercept

(-2,0)

, y-intercept

(0,-4)

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Setting $y=0$ gives $4x=8$ , so the x-intercept is $(2,0)$ . Setting $x=0$ gives $-2y=8$ , so $y=-4$ , producing $(0,-4)$ . Therefore neither sign nor coordinate should be reversed.

Q5. A school uses the linear model $6x+3y=30$ , where $x$ and $y$ represent two quantities. The manager wants to know how much of one quantity is possible when the other is zero. Which intercept pair provides this information?

(5,0)

and

(0,10)

✅

(10,0)

and

(0,5)

(5,0)

and

(0,-10)

(-5,0)

and

(0,10)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: When $y=0$ , the equation becomes $6x=30$ , giving $x=5$ . When $x=0$ , it becomes $3y=30$ , giving $y=10$ . These intercepts represent the model's endpoint values when one quantity is reduced to zero.

Q6. A student claims that for $7x+5y=35$ , the intercepts are $(7,5)$ because the coefficients are 7 and 5. How should this reasoning be corrected?

A.The coefficients are always the coordinates of the intercepts.

B.The intercepts are found by making one variable zero at a time, producing

(5,0)

and

(0,7)

. ✅

C.The intercepts are

(7,0)

and

(5,0)

D.The equation has no intercepts because both coefficients are positive.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The coefficients alone are not the intercept coordinates. Setting $y=0$ gives $7x=35$ , so $x=5$ . Setting $x=0$ gives $5y=35$ , so $y=7$ . The intercepts therefore require division by the relevant coefficient.

Q7. A budget equation is $8x+4y=40$ . A planner interprets the x-intercept as the maximum possible value of $x$ when $y=0$ . What is that maximum value?

A.4

B.5 ✅

C.8

D.10

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: At the x-intercept, $y=0$ , so the equation becomes $8x=40$ . Dividing by 8 gives $x=5$ . The intercept therefore represents the maximum modeled value of $x$ when the other quantity is zero, assuming the variables are restricted to nonnegative values.

Q8. A student uses $3x-6y=12$ and computes the y-intercept as $(0,2)$ . Which step reveals the error?

A.They should set

y=0

when finding the y-intercept.

B.They should set

x=0

, which gives

-6y=12

, so the y-intercept is

(0,-2)

. ✅

C.They should divide both coordinates by 6.

D.They should change the equation to

3x+6y=12

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For a y-intercept, $x$ must equal zero. Substitution produces $-6y=12$ , hence $y=-2$ . The student's positive value ignores the negative coefficient of $y$ . This is a common sign error caused by solving without preserving the equation's structure.

Q9. A line is represented by $2x+4y=16$ . On a coordinate grid, the line crosses the positive x-axis at one point and the positive y-axis at another. Which description matches those crossings?

A.It crosses at

x=4

and

y=8

B.It crosses the x-axis at

4

and the y-axis at

4

C.It crosses the x-axis at

8

and the y-axis at

4

. ✅

D.It crosses the x-axis at

16

and the y-axis at

16

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The x-axis crossing requires $y=0$ , giving $2x=16$ , so $x=8$ . The y-axis crossing requires $x=0$ , giving $4y=16$ , so $y=4$ . Thus the graph crosses at $(8,0)$ and $(0,4)$ .

Q10. Two equations are $x+2y=12$ and $3x+2y=18$ . Which comparison of their intercepts is correct?

A.They have identical x- and y-intercepts.

B.The first has x-intercept 12 and y-intercept 6; the second has x-intercept 6 and y-intercept 9. ✅

C.The first has x-intercept 6 and y-intercept 12; the second has x-intercept 9 and y-intercept 6.

D.The first has x-intercept 12 and y-intercept 2; the second has x-intercept 18 and y-intercept 2.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For $x+2y=12$ , setting $y=0$ gives $x=12$ , while setting $x=0$ gives $y=6$ . For $3x+2y=18$ , the corresponding values are $x=6$ and $y=9$ . Comparing both equations requires calculating each intercept independently.

Q11. A graph shows a straight line crossing the x-axis at $(6,0)$ and the y-axis at $(0,-3)$ . Which equation could represent the line?

x+2y=6

x-2y=6

✅

2x-y=12

3x+6y=18

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The intercepts determine the slope as $(-3-0)/(0-6)=1/2$ . A line through $(6,0)$ with slope $1/2$ has equation $y=(1/2)x-3$ . Rearranging gives $x-2y=6$ , matching option B.

Q12. A production model is $9x+6y=54$ . A manager wants to compare the maximum $x$ -only plan with the maximum $y$ -only plan. What is the ratio of the x-intercept value to the y-intercept value?

2:3

✅

3:2

1:2

2:1

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Setting $y=0$ gives $x=6$ , while setting $x=0$ gives $y=9$ . Therefore the ratio of the x-intercept value to the y-intercept value is $6:9$ , which simplifies to $2:3$ . This requires finding both intercepts before comparing them.

Q13. For $12x+8y=48$ , a student argues that the intercepts are $(4,0)$ and $(0,6)$ , but another student says the intercepts should be $(48,0)$ and $(0,48)$ because the constant is 48. Which conclusion is mathematically justified?

A.The first student is correct because each intercept requires setting the other variable to zero. ✅

B.The second student is correct because the constant always equals both intercept values.

C.Both students are correct under different graphing conventions.

D.Neither student is correct because intercepts cannot be found from a standard equation.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Setting $y=0$ gives $12x=48$ , so $x=4$ . Setting $x=0$ gives $8y=48$ , so $y=6$ . The constant $48$ is not itself an intercept; it must be divided by the coefficient of the remaining variable after the other variable is set to zero.

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