📝 Graphing linear equations by plotting points (16 MCQs)

📖 From Digital SAT Algebra • 4. Graphs • 16 questions available

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What is Graphing linear equations by plotting points?

Definition:
Graphing linear equations by plotting points is a method where we select a set of x-values, compute the corresponding y-values from the equation, create ordered pairs, plot these points on the coordinate plane, and then draw a straight line through them; this method is straightforward and works for any linear equation, providing a visual representation of the equation's solutions and the relationship between the variables.

Working:
First, choose a few x-values (e.g., -2, 0, 2), substitute each into the equation (e.g., $y = 3x - 1$ ) to find y, list the pairs in a table, plot the points on the coordinate plane, and if they are collinear, draw a line through them; it is important to choose x-values that are easy to compute and spread out to ensure accuracy; for example, for $y = 2x - 3$ , using x = -1, 0, 1 gives y = -5, -3, -1, and the points (-1, -5), (0, -3), (1, -1) are plotted to draw the line.

Example:
A simple example is $y = \frac{2}{3}x + 1$ ; choose x = -3, 0, 3, then y = -1, 1, 3, giving points (-3, -1), (0, 1), (3, 3); plotting these points and drawing a line graphs the equation, illustrating the method.

Reason:
Plotting points is the most fundamental graphing technique, building the concept of graphing from scratch, and it is useful for students to understand how equations translate into lines, making it an essential skill in algebra and precalculus.

Easy

Medium

Hard

📝 All Graphing linear equations by plotting points MCQs

Q1. A student wants to graph $y=2x-3$ by plotting points. Which pair of points is sufficient to determine the line correctly?

(0,-3)

and

(2,1)

✅

(0,3)

and

(2,1)

(-1,-1)

and

(1,3)

(1,-3)

and

(2,-1)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: For $x=0$ , the equation gives $y=-3$ , producing $(0,-3)$ . For $x=2$ , it gives $y=1$ , producing $(2,1)$ . Since two distinct points determine a linear graph, this pair is sufficient.

Q2. When graphing $y=-3x+4$ , a student chooses $x=1$ and $x=2$ . Which coordinates should be plotted?

(1,1)

and

(2,-2)

✅

(1,7)

and

(2,10)

(1,-1)

and

(2,-2)

(1,4)

and

(2,1)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Substituting $x=1$ gives $y=-3(1)+4=1$ , while $x=2$ gives $y=-6+4=-2$ . Therefore, the correct points are $(1,1)$ and $(2,-2)$ , which reflect the negative slope.

Q3. Two students graph $y=x+2$ . Student A uses $x=0,1$ , while Student B uses $x=-5,5$ . Which conclusion is most accurate?

A.Only Student A can graph the equation

B.Only Student B can graph the equation

C.Both methods can determine the same line if the coordinates are calculated correctly ✅

D.Neither method works because three points are required

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Any two distinct points satisfying the same linear equation determine the same straight line. Student A obtains $(0,2)$ and $(1,3)$ , while Student B obtains $(-5,-3)$ and $(5,7)$ . Both methods are valid.

Q4. A graphing task requires plotting $y=\frac{1}{2}x-4$ . Which choice makes the plotting process easiest while reducing the chance of fractional coordinates?

A.Use

x=1

and

x=3

B.Use

x=0

and

x=2

✅

C.Use

x=-1

and

x=1

D.Use

x=4

and

x=6

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Choosing $x=0$ and $x=2$ produces $y=-4$ and $y=-3$ , both integers. This avoids fractional coordinates and makes accurate plotting easier, while still providing two distinct points needed for the line.

Q5. A school models the cost of notebooks using $y=50x+100$ , where $x$ represents the number of notebooks. Which plotted point correctly represents buying 4 notebooks?

(4,150)

(4,200)

(4,300)

✅

(4,350)

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Substituting $x=4$ gives $y=50(4)+100=200+100=300$ . Thus $(4,300)$ represents four notebooks and a total modeled cost of 300 units. The other choices result from incorrect multiplication or ignoring the fixed cost.

Q6. A student plots $(-2,5)$ and $(2,-3)$ for a line. Without calculating the equation first, what feature can be inferred from these points?

A.The line has positive slope

B.The line has negative slope ✅

C.The line must pass through the origin

D.The line must be horizontal

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: As $x$ increases from $-2$ to $2$ , $y$ decreases from $5$ to $-3$ . A decrease in $y$ as $x$ increases indicates a negative slope, so the line falls from left to right.

Q7. A delivery company uses $y=25x+75$ , where $x$ is the number of deliveries and $y$ is total earnings. A graph is created using $x=0,2,4$ . Which sequence of points should appear?

(0,75),(2,125),(4,175)

✅

(0,25),(2,75),(4,125)

(0,75),(2,100),(4,125)

(0,75),(2,150),(4,250)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Substitution gives $y=75$ when $x=0$ , $y=125$ when $x=2$ , and $y=175$ when $x=4$ . Therefore, the correct sequence is $(0,75),(2,125),(4,175)$ , showing a constant increase.

Q8. A student claims that plotting $(1,4)$ and $(3,8)$ is enough to graph $y=2x+2$ , but a classmate says three points are always required. Who is correct?

A.The student is correct because two distinct valid points determine a linear graph ✅

B.The classmate is correct because every line requires three points

C.Both are wrong because four points are required

D.The student is wrong because one point is enough

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Checking the equation gives $4$ when $x=1$ and $8$ when $x=3$ , so both plotted points are valid. Two distinct points uniquely determine a straight line, making a third point useful only as a verification.

Q9. A student graphs $y=-2x+6$ using the points $(0,6)$ and $(3,0)$ . Another student says the second point should be $(3,12)$ . What is the error?

A.The student reversed the coordinates

B.The student multiplied

3

6

instead of substituting into the equation ✅

C.The student used the wrong sign for the slope

D.The student forgot that

x

must be negative

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: When $x=3$ , substitution gives $y=-2(3)+6=-6+6=0$ , so $(3,0)$ is correct. The incorrect value $12$ likely comes from multiplying $3$ by $6$ without applying the equation.

Q10. A student plots $(-1,2)$ , $(0,4)$ , and $(1,6)$ for a linear equation. Which equation is most consistent with these points?

y=x+4

y=2x+4

✅

y=2x-4

y=-2x+4

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The points show that increasing $x$ by 1 increases $y$ by 2, so the slope is $2$ . When $x=0$ , $y=4$ , giving the equation $y=2x+4$ , which satisfies all three plotted points.

Q11. A graph contains the plotted points $(0,-2)$ , $(2,2)$ , and $(4,6)$ , connected by a straight line. Which equation matches the graph?

y=x-2

y=2x-2

✅

y=2x+2

y=-2x-2

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: From $(0,-2)$ to $(2,2)$ , $y$ increases by 4 while $x$ increases by 2, giving slope $2$ . The point $(0,-2)$ gives the vertical intercept, so the equation is $y=2x-2$ .

Q12. A line is graphed from two points $(-3,7)$ and $(1,-1)$ . A student says the line rises as it moves from left to right. What should the student conclude instead?

A.The line rises because both coordinates increase

B.The line falls because

y

decreases while

x

increases ✅

C.The line is horizontal because two points were used

D.The line cannot be classified without a third point

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Moving from $(-3,7)$ to $(1,-1)$ , the $x$ -coordinate increases while the $y$ -coordinate decreases. Therefore, the line falls from left to right, indicating a negative slope rather than a positive one.

Q13. A water tank model is $y=120-15x$ , where $x$ is hours after monitoring begins and $y$ is liters remaining. Which pair of points is most useful for plotting the model?

(0,120)

and

(4,60)

✅

(0,15)

and

(4,120)

(1,120)

and

(4,15)

(0,-120)

and

(4,-60)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: At $x=0$ , the model gives $y=120$ , and at $x=4$ , it gives $y=120-60=60$ . These two points are easy to calculate and plot, and they clearly show the model's decreasing trend.

Q14. Two students graph $y=3x-1$ . Student A makes a table using $x=-2,0,2$ . Student B chooses $x=1,2,3$ . Which method is better for checking whether the graph was plotted correctly?

A.Student A, because negative and positive

x

-values provide a wider check ✅

B.Student B, because positive values are always more accurate

C.Student A, because only negative values should be used

D.Both methods are invalid unless

x=0

is excluded

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Student A's choices produce $y=-7,-1,5$ , covering negative, zero, and positive $x$ -values. This broader range helps reveal plotting or calculation errors and makes it easier to check whether the resulting points form the expected straight line.

Q15. A student correctly calculates points for $y=-x+5$ but places $(2,3)$ at $(3,2)$ on the coordinate plane. What is the most likely consequence?

A.The line will still be correct because the coordinates are equivalent

B.The plotted line may have the wrong slope because one coordinate pair was misplaced ✅

C.Only the vertical intercept will change

D.The graph will become horizontal

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The ordered pair $(2,3)$ means move 2 units horizontally and 3 units vertically. Plotting it as $(3,2)$ changes both coordinates, creating an incorrect point. A line through that mistaken point can have a different slope and intercept.

Q16. For $y=4x-8$ , a student wants to choose two points that make the graph especially easy to verify. Which pair provides the strongest check?

(0,-8)

and

(2,0)

✅

(1,-4)

and

(2,-8)

(0,8)

and

(2,16)

(-2,0)

and

(0,8)

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Substitution gives $(0,-8)$ and $(2,0)$ , both simple integer points. They also show where the line crosses each axis, making the plotted graph easier to verify. The other pairs contain sign or substitution errors.

🔗 Related Topics (MCQs)

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