📝 Graphing linear equations in two variables (17 MCQs)

📖 From Digital SAT Algebra • 4. Graphs • 17 questions available

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What is Graphing linear equations in two variables?

Definition:
Graphing linear equations in two variables involves plotting the set of all solutions (x, y) on the coordinate plane, which forms a straight line, and this is achieved by finding at least two points that satisfy the equation, plotting them, and drawing a line through them; the graph visually represents the relationship between the two variables, and it is a powerful tool for analyzing and solving problems.

Working:
To graph a linear equation in the form $y = mx + b$ , plot the y-intercept (0, b), then use the slope to find another point, or make a table of values, plot the points, and draw the line; if the equation is in standard form $Ax + By = C$ , find the x- and y-intercepts and plot them; for example, for $y = 2x + 1$ , plot (0, 1) and (1, 3), then draw the line; the graph extends infinitely in both directions, and every point on the line is a solution, representing all possible pairs.

Example:
A simple example is graphing $y = -x + 4$ ; plot the y-intercept (0, 4), then use the slope -1 to find another point (1, 3), or plot (0, 4) and (2, 2), draw a line through them; another example is graphing $x - 2y = 4$ , by finding intercepts: if x=0, y=-2, giving (0, -2); if y=0, x=4, giving (4, 0); plotting these and drawing the line graphs the equation.

Reason:
Graphing is a visual method that makes relationships intuitive, helps solve systems of equations, and is essential in many fields, including physics, economics, and engineering, making it a critical skill in algebra and applied mathematics.

Easy

Medium

Hard

📝 All Graphing linear equations in two variables MCQs

Q1. Which pair of points is sufficient to graph the linear equation $y=2x-3$ accurately?

(0, -3)

and

(1, -1)

✅

(0, 3)

and

(1, 5)

(-1, -1)

and

(1, 1)

(1, -3)

and

(2, -1)

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: A linear equation can be graphed using any two distinct solutions. Substituting $x=0$ gives $y=-3$ , and substituting $x=1$ gives $y=-1$ . Therefore, both points satisfy the equation and determine the required straight line.

Q2. A student wants to graph $3x+y=6$ . Which first step is most useful for producing a graph efficiently?

A.Rewrite it as

y=6-3x

✅

B.Rewrite it as

x=6-3y

C.Set

x=0

only

D.Set

y=0

only

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Rewriting the equation as $y=6-3x$ places it in a form where the starting value and rate of change are immediately visible. This makes selecting points and plotting the straight line much more efficient.

Q3. Two students graph $y=-2x+4$ . Student A uses the points $(0,4)$ and $(2,0)$ , while Student B uses $(1,2)$ and $(3,-2)$ . What is the best conclusion?

A.Only Student A is correct

B.Only Student B is correct

C.Both students produce the same line ✅

D.Neither student can graph the equation

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Both sets contain valid solutions of $y=-2x+4$ . The first pair represents the intercepts, while the second pair represents other points on the same line. Any two distinct correct solutions determine the same linear graph.

Q4. A graph passes through $(0,-2)$ and rises 3 units for every 1-unit increase in $x$ . Which equation represents the graph?

y=3x-2

✅

y=-3x-2

y=3x+2

y=-2x+3

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The point $(0,-2)$ shows that the vertical intercept is $-2$ . A rise of 3 for every run of 1 gives a slope of 3. Combining these features produces $y=3x-2$ .

Q5. Why can a graph of $2x-4y=8$ be constructed from only two correctly plotted points?

A.Because every equation has exactly two solutions

B.Because a linear equation produces a straight line, and two distinct points determine that line ✅

C.Because the graph must pass through the origin

D.Because only intercepts can be used

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A linear equation in two variables represents a straight line when it has the usual nondegenerate form. Once two distinct points on that line are known, the line is uniquely determined, so additional points are unnecessary for constructing it.

Q6. A student claims that the equation $y=-\frac{1}{2}x+5$ should be graphed by moving 1 unit right and 2 units up from $(0,5)$ . What is wrong with the reasoning?

A.The intercept should be negative

B.The slope is negative, so moving right requires moving 2 units down ✅

C.The graph cannot use

(0,5)

D.The slope should be 2 instead of

-\frac12

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The slope $-\frac12$ means that for every 1-unit increase in $x$ , $y$ decreases by $\frac12$ . Equivalently, moving 2 units right requires moving 1 unit down. The student's movement reverses the sign and changes the slope.

Q7. A delivery service charges a fixed fee of \ $8 plus \$ 3 for every kilometer traveled. Which graph feature identifies the fixed fee?

A.The slope

B.The horizontal intercept

C.The vertical intercept ✅

D.The direction of the

x

-axis

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The model is $y=3x+8$ , where $x$ represents kilometers and $y$ represents total cost. When $x=0$ , the cost is \$8, so the graph crosses the vertical axis at $(0,8)$ , representing the fixed fee.

Q8. A water tank contains 500 liters and loses 20 liters each hour. Which equation and graph description correctly model the amount of water remaining?

y=500+20x

, increasing line

y=500-20x

, decreasing line ✅

y=20x-500

, increasing line

y=-500-20x

, decreasing line

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The initial amount is 500 liters, giving a vertical intercept of 500. Since the tank loses 20 liters each hour, the rate is $-20$ . Therefore, $y=500-20x$ , and the graph decreases as time increases.

Q9. A student plots $(0,6)$ and $(2,2)$ for $y=-2x+6$ , but the points appear visually correct. How can the student verify the graph without relying on appearance?

A.Check whether both coordinates satisfy the equation ✅

B.Check whether both points have positive coordinates

C.Check whether the line passes through the origin

D.Check whether the slope is positive

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Visual placement can hide plotting errors. Substitution provides a reliable check: $0,6$ gives $6=-2(0)+6$ , and $2,2$ gives $2=-2(2)+6$ . Both satisfy the equation, confirming the plotted points.

Q10. A graph of $y=\frac{3}{4}x-2$ is being constructed. Which pair of moves from $(0,-2)$ will reach another exact point on the line?

A.4 units right and 3 units up ✅

B.4 units right and 3 units down

C.3 units right and 4 units up

D.3 units left and 4 units up

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The slope is $\frac34$ , meaning rise 3 for a run of 4. Starting from $(0,-2)$ , moving 4 units right and 3 units up gives another solution. This approach avoids decimal coordinates and makes accurate plotting easier.

Q11. Two lines are represented by $y=2x+1$ and $y=2x-4$ . Without plotting many points, what should a student predict about their graphs?

A.They intersect at the origin

B.They are parallel because they have equal slopes but different intercepts ✅

C.They are identical because their slopes match

D.They intersect at

(0,1)

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Both equations have slope 2, so the lines have the same direction. Their vertical intercepts are different, 1 and $-4$ , so they are distinct rather than identical. Therefore, the graphs are parallel and do not intersect.

Q12. A line passes through $(-2,7)$ and $(4,-5)$ . Which equation should result if the points are used correctly?

y=-2x+3

✅

y=2x+3

y=-2x-3

y=3x-2

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The slope is $(-5-7)/(4-(-2))=-12/6=-2$ . Using $(-2,7)$ , $7=-2(-2)+b$ , so $b=3$ . Thus the equation is $y=-2x+3$ , and both original points satisfy it.

Q13. A line representing a taxi fare starts at \ $50 when the distance is zero and increases by \$ 12 per kilometer. A student draws a line with vertical intercept 12 and slope 50. Which error occurred?

A.The axes were reversed

B.The starting value and rate were interchanged ✅

C.The slope should be negative

D.The graph should pass through the origin

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The fixed starting fare is represented by the vertical intercept, so it must be 50. The \$12 increase per kilometer is the slope. Interchanging these values produces an incorrect model even though both numbers appear in the student's equation.

Q14. A graph contains the points $(-3,5)$ , $(0,2)$ , and $(3,-1)$ . Which conclusion best demonstrates that these points lie on one linear graph?

A.The

x

-coordinates are equally spaced

B.The

y

-coordinates are all positive

C.The change in

y

is consistently

-3

when

x

increases by 3 ✅

D.The graph contains three points

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: From $(-3,5)$ to $(0,2)$ , $x$ increases by 3 while $y$ decreases by 3. From $(0,2)$ to $(3,-1)$ , the same changes occur. Consistent rate of change confirms collinearity.

Q15. A student graphs $x+y=4$ using $(0,4)$ and $(4,0)$ . Another student graphs it using $(1,3)$ and $(3,1)$ . If both graphs are drawn accurately, what must happen?

A.The lines have different slopes

B.The lines overlap because all four points satisfy the same equation ✅

C.Only the first graph represents the equation

D.The second graph must be curved

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Each of the four points satisfies $x+y=4$ . Since the equation represents one straight line, both pairs of points determine that same line. The students use different points, but their correctly drawn graphs must coincide.

Q16. A line has equation $y=mx+b$ . It passes through $(2,7)$ , and its vertical intercept is $-1$ . Which reasoning correctly determines its graph?

A.The slope is

m=3

, so plot

(0,-1)

and

(2,7)

B.The slope is

m=4

, so plot

(0,-1)

and

(2,7)

✅

C.The slope is

m=-4

, so plot

(0,-1)

and

(2,7)

D.The slope is

m=8

, so plot

(0,-1)

and

(2,7)

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The vertical intercept gives $b=-1$ . Using the two points $(0,-1)$ and $(2,7)$ , the slope is $(7-(-1))/(2-0)=8/2=4$ . Therefore, the graph is $y=4x-1$ .

Q17. A line through integer-coordinate points has a slope of $-\frac{2}{3}$ and crosses the vertical axis at 5. A second line has slope $-\frac{2}{3}$ and crosses the vertical axis at 5 as well. What is the strongest conclusion?

A.The lines intersect once

B.The lines are perpendicular

C.The lines are the same line ✅

D.The lines have no common points

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Two linear equations with the same slope and the same vertical intercept represent the same set of points. Therefore, the graphs coincide completely rather than merely being parallel or intersecting at one point.

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