📝 Graphing linear equations using intercepts (13 MCQs)

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

←Previous Topic:How to find x and y intercepts from equation📝 Graphing linear equations using interceptsSlope of a line definition:Next Topic→

What is Graphing linear equations using intercepts?

Definition:
Graphing linear equations using intercepts is a method where we use the x-intercept (where y=0) and the y-intercept (where x=0) to plot the line, as these two points are sufficient to determine a straight line; this technique is efficient, especially for equations in standard form, and it provides a clear geometric representation of the equation's solutions.

Working:
First, find the x-intercept by setting y=0 and solving for x, and find the y-intercept by setting x=0 and solving for y; then, plot these two points on the coordinate plane and draw a straight line through them; for example, for $2x + y = 4$ , x-intercept: set y=0, $2x = 4 \Rightarrow x = 2$ , point (2,0); y-intercept: set x=0, $y = 4$ , point (0,4); plot these and draw the line; this method works for any line that does not pass through the origin (where intercepts are the same point), but if both intercepts are at (0,0), use another point.

Example:
A simple example is $x + 3y = 6$ ; x-intercept: set y=0, x = 6, so (6,0); y-intercept: set x=0, 3y = 6 \Rightarrow y = 2, so (0,2); plot these and draw the line, illustrating the intercept method.

Reason:
Graphing using intercepts is a quick and efficient method, emphasizing key points and reducing computation, making it a preferred method in many applications and a fundamental skill for students of algebra.

Easy

Medium

Hard

📝 All Graphing linear equations using intercepts MCQs

Q1. For the equation $3x+2y=12$ , which pair of points is sufficient to graph the line using the intercept method?

A.The points

(4,0)

and

(0,6)

✅

B.The points

(3,0)

and

(0,4)

C.The points

(12,0)

and

(0,12)

D.The points

(0,4)

and

(4,0)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: To use intercepts, set $y=0$ to obtain $x=4$ , giving $(4,0)$ , and set $x=0$ to obtain $y=6$ , giving $(0,6)$ . These two intercepts determine the line.

Q2. A student says that the equation $x+5y=10$ has intercepts $(10,0)$ and $(0,10)$ . What is the best evaluation of the student's reasoning?

A.Both intercepts are correct because the coefficients determine the coordinates

B.Only the

x

-intercept is correct; the

y

-intercept should be

(0,2)

✅

C.Only the

y

-intercept is correct; the

x

-intercept should be

(5,0)

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

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Setting $y=0$ gives $x=10$ , so $(10,0)$ is correct. Setting $x=0$ gives $5y=10$ , or $y=2$ , so the correct $y$ -intercept is $(0,2)$ .

Q3. Why does finding the two intercepts usually provide enough information to graph a nonvertical, nonhorizontal line?

A.The intercepts identify two distinct points that determine one unique line ✅

B.The intercepts automatically give the slope without any calculation

C.The intercepts guarantee that the line passes through the origin

D.The intercepts show every possible solution of the equation

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: A nonvertical, nonhorizontal line is uniquely determined by two distinct points. The $x$ - and $y$ -intercepts provide those points, allowing the line to be drawn without calculating additional coordinates.

Q4. Which equation would produce an $x$ -intercept of $(6,0)$ and a $y$ -intercept of $(0,3)$ ?

x+2y=6

3x+6y=18

✅

x+2y=12

2x+y=12

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For $3x+6y=18$ , setting $y=0$ gives $x=6$ , while setting $x=0$ gives $y=3$ . Therefore, both stated intercepts are produced by this equation.

Q5. A delivery company models a cost relationship with $4x+8y=32$ , where $x$ and $y$ represent two types of units. What do the intercepts mean in this model?

A.They represent combinations where both unit types are used equally

B.They represent the maximum values of

x

and

y

when the other variable is zero ✅

C.They represent the slope and

y

-intercept of the model

D.They represent points where the cost becomes negative

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Setting $y=0$ gives $x=8$ , while setting $x=0$ gives $y=4$ . Each intercept represents the maximum amount of one unit type when none of the other type is used.

Q6. A line has intercepts $(8,0)$ and $(0,4)$ . A student graphs $(8,4)$ and $(0,0)$ instead. What misconception most directly caused the error?

A.Confusing the intercepts with their coordinate values and combining them incorrectly ✅

B.Assuming every line must have a positive slope

C.Using too many points to determine a line

D.Switching the

x

- and

y

-axes

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The intercepts are separate points: $(8,0)$ lies on the $x$ -axis and $(0,4)$ lies on the $y$ -axis. Combining the nonzero coordinates creates $(8,4)$ , which is not an intercept.

Q7. A student wants to graph $6x-3y=18$ . They calculate the $x$ -intercept as $(3,0)$ and the $y$ -intercept as $(0,-6)$ . Which conclusion is correct?

A.Both calculations are correct ✅

B.Only the

x

-intercept is correct

C.Only the

y

-intercept is correct

D.Both calculations are incorrect

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Setting $y=0$ gives $6x=18$ , so $x=3$ . Setting $x=0$ gives $-3y=18$ , so $y=-6$ . Thus the student's two intercepts are correct.

Q8. A rectangular garden has a boundary modeled by $2x+3y=12$ . If only nonnegative values are meaningful, which intercept pair represents the endpoints of the portion of the boundary inside the first quadrant?

(6,0)

and

(0,4)

✅

(12,0)

and

(0,12)

(4,0)

and

(0,6)

(-6,0)

and

(0,-4)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: For the $x$ -intercept, $y=0$ gives $2x=12$ , so $x=6$ . For the $y$ -intercept, $x=0$ gives $3y=12$ , so $y=4$ . Both satisfy the nonnegative restriction.

Q9. Two students graph $5x+2y=20$ . Student A finds intercepts $(4,0)$ and $(0,10)$ . Student B finds $(20,0)$ and $(0,20)$ . Which student's graph can represent the equation?

A.Only Student A's graph ✅

B.Only Student B's graph

C.Both graphs represent the same line

D.Neither graph represents the equation

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: For $5x+2y=20$ , setting $y=0$ gives $x=4$ , and setting $x=0$ gives $y=10$ . Student B incorrectly assumes the constant is automatically both intercept values.

Q10. A graph shows a line crossing the axes at $(3,0)$ and $(0,-6)$ . Which statement correctly describes what should happen when the line is graphed using intercepts?

A.The line should pass through the origin

B.The line should connect the two intercepts and extend in both directions ✅

C.The line should remain only between the two intercepts

D.The line should be horizontal because one intercept is negative

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The points $(3,0)$ and $(0,-6)$ are two points on the same line. Plotting them and extending the resulting straight line in both directions produces the complete graph.

Q11. A student finds the intercepts of $2x+4y=16$ as $(8,0)$ and $(0,4)$ , then claims the line rises from left to right. What should be concluded?

A.The claim is correct because both intercepts are positive

B.The claim is incorrect because the line has negative slope ✅

C.The claim is correct because the

y

-intercept is positive

D.The claim cannot be determined from intercepts

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Rewriting the equation gives $y=4-\frac{1}{2}x$ , so the slope is negative. As $x$ increases, $y$ decreases, meaning the line falls from left to right despite both intercepts being positive.

Q12. Suppose a line has $x$ -intercept $(12,0)$ and $y$ -intercept $(0,8)$ . Which equation can be obtained directly from these intercepts?

2x+3y=24

✅

3x+2y=24

8x+12y=96

x+y=20

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Using the intercept form gives $\frac{x}{12}+\frac{y}{8}=1$ . Multiplying by $24$ produces $2x+3y=24$ . This equation correctly yields both stated intercepts when the other variable is zero.

Q13. A graphing mistake produces intercepts $(0,7)$ and $(7,0)$ for the equation $x+y=14$ . Without fully redrawing the graph, how can the error be detected?

A.Substitute either claimed intercept into the equation; each gives

7

, not

14

✅

B.The points have equal coordinates, so they cannot be intercepts

C.An intercept must always be negative

D.The equation cannot have two intercepts

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Testing $(7,0)$ gives $7+0=7$ , and testing $(0,7)$ gives $0+7=7$ , neither equals $14$ . The correct intercepts are $(14,0)$ and $(0,14)$ .

🔗 Related Topics (MCQs)

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