Question 1

Passage

A student is given two points on a line: (0, 2) and (3, 8). The student states, “The slope is 8 − 2 = 6."

Question

Which statement best identifies the error in the student’s reasoning?

Accepted Answer

The student did not divide the change in y by the change in x.

Answer

The student must find the y-intercept before computing slope.

Answer

The student should subtract the x-values only.

Answer

The student’s work is correct.

Question 2

Passage

To find the slope through $\left(-2,5\right)$ and $\left(4,-1\right)$ , a student writes: $m=\frac{-1-5}{-2-4}=\frac{-6}{-6}=1$ .

Question

What is the primary flaw in the student's computation?

Accepted Answer

The student subtracted y-values and x-values using inconsistent point order.

Answer

The student should have added the coordinates instead of subtracting.

Answer

The student should have converted the points into intercept form first.

Answer

The student incorrectly simplified a negative fraction.

Question 3

Passage

A student rewrites $\left(-2,5\right)$ as : $\left(4,-1\right)$

Question

What is the flaw in the student's algebra?

Accepted Answer

The student divided by

-2

incorrectly and lost the negative sign.

Answer

The student moved $3x$ incorrectly (it should become

\left(-2,5\right)

).

Answer

Standard form cannot be rewritten in slope-intercept form.

Answer

The student should have solved for $3x$ instead of $\left(-2,5\right)$ .

Question 4

Passage

A student examines $y=-4x+7$ and claims, “The y-intercept is −4.”

Question

What is the student confusing?

Accepted Answer

The slope with the y-intercept.

Answer

The coefficient with the exponent

Answer

The x-intercept with the y-intercept

Answer

The y-intercept with the domain

Question 5

Passage

To find the x-intercept of $\left(-2,5\right)$ , a student sets

\left(4,-1\right)

, obtains

m=\frac{-1-5}{-2-4}=\frac{-6}{-6}=1

, and concludes the x-intercept is −6.

Question

What is the error in this approach?

Accepted Answer

The student should set $3x$ to find the x-intercept.

Answer

The student should set

-2

to find the x-intercept.

Answer

The student should set $3x$ to find the x-intercept.

Answer

Intercepts cannot be found algebraically.

Question 6

Passage

A line has slope 5 and passes through (0, −3). A student writes the equation $y=-4x+7$ .

Question

Which statement best explains the mistake?

Accepted Answer

The student used the correct intercept value but the wrong sign.

Answer

The student assumed the line must pass through (3, 0)

Answer

The student used the x-intercept instead of the y-intercept.

Answer

The student inverted the slope.

Question 7

Passage

Using points (1, 9) and (3, 3), a student computes:

y=-4x+7

Question

What is the flaw in the slope calculation?

Accepted Answer

The student subtracted y-values and x-values from different point orders.

Answer

The student should use the midpoint formula first.

Answer

The student should compute $3x$ instead.

Answer

The student must graph the line before finding slope.

Question 8

Passage

A table shows:

x: 2, 4, 6

y: 10, 4, −2

A student says, “The slope is 3 because y changes by 6 each time.”

Question

What is the most accurate critique?

Accepted Answer

The student ignored that the change in y is negative, not positive.

Answer

The student used the intercept instead of slope

Answer

The student should use only the first two points.

Answer

Tables cannot represent linear functions

Question 9

Passage

A student rewrites $y=-4x+7$ as : $y-4=-2x+1\ \Rightarrow\ y=-2x+5$

Question

Where did the student’s reasoning break down?

Accepted Answer

The student distributed −2 incorrectly to the +1.

Answer

The student subtracted 4 instead of adding 4.

Answer

The student should have divided both sides by −2 first.

Answer

The equation cannot be written in slope-intercept form.

Question 10

Passage

A student sees $y=-4x+7$ and says, “The y-intercept is −7.”

Question

What is the best explanation of the student’s error?

Accepted Answer

A vertical line has no y-intercept unless it is $3x$ .

Answer

Vertical lines always have slope 0.

Answer

Intercepts cannot be negative

Answer

The y-intercept is always the constant in the equation.

Question 11

Passage

Given $y=-4x+7$ , a student states, “The slope is $y-4=-2x+1\ \Rightarrow\ y=-2x+5$ because you flip the fraction."

Question

What is wrong with this reasoning?

Accepted Answer

The student incorrectly took the reciprocal of the slope.

Answer

Fractional slopes are not permitted.

Answer

The student confused slope with the y-intercept.

Answer

The student must compute slope using two points only.

Question 12

Passage

A student rewrites $y=-4x+7$ as : $y-4=-2x+1\ \Rightarrow\ y=-2x+5$

Question

What is the correct diagnosis?

Accepted Answer

The student divided by 2 instead of dividing by 5.

Answer

The student should have found intercepts first.

Answer

The student divided only part of the expression by 5.

Answer

The student moved $3x$ incorrectly.

Question 13

Passage

A line crosses the y-axis at (0, 6) and the x-axis at (2, 0). A student computes: $y=-4x+7$

Question

What is the flaw in the student's slope computation?

Accepted Answer

The student reversed only the y-subtraction, creating an incorrect sign.

Answer

The student should subtract x-values first.

Answer

The slope must be undefined because intercepts were used.

Answer

Intercepts cannot be used to find slope.

Question 14

Passage

To find the x-intercept of $y=-4x+7$ , a student sets

y-4=-2x+1\ \Rightarrow\ y=-2x+5

and gets

y=1

, then claims “The x-intercept is (0, 1)."

Question

What did the student actually find?

Accepted Answer

The y-intercept

Answer

The midpoint

Answer

<P>The slope</P>

Answer

The x-coordinate

Question 15

Passage

A student expands

y=-4x+7

as

y-4=-2x+1\ \Rightarrow\ y=-2x+5

.

Question

What is the student’s error?

Accepted Answer

The student failed to distribute 4 to the −2

Answer

The student incorrectly combined like terms

Answer

The student changed the slope by simplifying

Answer

The student should have divided by 4 before distributing

Question 16

Passage

A student finds the slope through (−3, −2) and (1, 6) by writing:

y=-4x+7

.

Question

What is the flaw?

Accepted Answer

The student used 3 instead of −3 in the x-subtraction.

Answer

The student subtracted y-values in the wrong order only.

Answer

Negative slopes are not possible.

Answer

The student should have added 3 instead of subtracting.

Question 17

Passage

A line has slope −3 and passes through (2, 5). A student writes:

y=-4x+7

So the student concludes

y-4=-2x+1\ \Rightarrow\ y=-2x+5

.

Question

What is the student’s mistake?

Accepted Answer

They solved for $3x$ incorrectly. (sign/operation error)

Answer

A line with slope - 3 cannot pass through (2, 5).

Answer

They substituted the point incorrectly.

Answer

They should have used point-slope form only

Question 18

Passage

A student says: “In $y=-4x+7$ , the y-intercept is −6.”

Question

What is wrong with the student’s conclusion?

Accepted Answer

They must isolate y before reading the y-intercept.

Answer

The x-intercept must be found first.

Answer

The slope cannot be determined from this equation.

Answer

The y-intercept can never be negative.

Question 19

Passage

A student says: “If the y-values in a table stay the same, then the slope is undefined.”

Question

What is the flaw in that statement?

Accepted Answer

Constant y-values imply slope 0, not undefined.

Answer

Undefined slope occurs only when y-values are constant.

Answer

Constant y-values imply slope 1.

Answer

Tables cannot show slope.

Question 20

Passage

A student reads: “The line rises 2 units for every 1 unit to the right,” and states, “So the slope is $y=-4x+7$ ."

Question

What is the student's error?

Accepted Answer

They reversed rise and run.

Answer

They treated the rate as an intercept.

Answer

They confused slope with y-intercept.

Answer

They assumed the line decreases.

Question 21

Passage

A student claims: “In $y=-4x+7$ , the slope is $y-4=-2x+1\ \Rightarrow\ y=-2x+5$ ."

Question

Is the student's interpretation correct?

Accepted Answer

No, the slope is $3x$ after solving for y.

Answer

Yes, because the slope is the constant term divided by $3x$ .

Answer

Yes, because the slope equals $3x$ in $\left(-2,5\right)$ .

Answer

No, the slope is $3x$ .

Question 22

Passage

A student states: “In $y=-4x+7$ , the value $y-4=-2x+1\ \Rightarrow\ y=-2x+5$ represents the x-intercept.”

Question

Is the student’s statement accurate?

Accepted Answer

No, $3x$ is the y-intercept.

Answer

Yes, because intercepts always come from constants.

Answer

No, $3x$ represents the slope.

Answer

No, $3x$ is unknown.

Question 23

Passage

A student says: “If the y-intercept is 0, then the line must be vertical.”

Question

Is this correct?

Accepted Answer

No, it means the line passes through the origin

Answer

Yes, because a y-intercept of 0 means the equation must be

-2

Answer

No, because a y-intercept of 0 forces the slope to be undefined

Answer

Yes, because the y-axis becomes the line

Question 24

Passage

A student claims: “The graph of $y=-4x+7$ has slope 0 and y-intercept 7.”

Question

Is the student correct?

Accepted Answer

Yes, the slope is 0 and y-intercept is 7.

Answer

No, the slope is 7.

Answer

No, horizontal lines have no y-intercept.

Answer

No, the slope is undefined.

Question 25

Passage

A student says: “For the equation $y=-4x+7$ , the slope is 3.”

Question

Is this interpretation correct?

Accepted Answer

No, vertical lines have undefined slope.

Answer

No, because vertical lines always have slope 0.

Answer

Yes, because the constant equals the slope.

Answer

Yes, the slope is 3.

Question 26

Passage

A student states: “If a line has slope 0, then it cannot cross the x-axis.”

Question

Is the statement always true?

Accepted Answer

No, because

-2

has slope 0 and lies on the x-axis

Answer

Yes, because any line with slope 0 stays above or below the x-axis

Answer

No, because slope 0 means the line is vertical

Answer

Yes, because slope 0 implies the line has no intercepts

Question 27

Passage

A student says: “For $y=-4x+7$ , the x-intercept is 4.”

Question

Is the student’s interpretation correct?

Accepted Answer

Yes, because setting

\left(-2,5\right)

gives

\left(4,-1\right)

, so the intercept is

m=\frac{-1-5}{-2-4}=\frac{-6}{-6}=1

Answer

No, because the x-intercept is 8

Answer

Yes, because the x-intercept is always equal to the y-intercept

Answer

No, because the x-intercept is $\left(-2,5\right)$

Question 28

Passage

A student claims: “In $y=-4x+7$ , the y-intercept is always

y-4=-2x+1\ \Rightarrow\ y=-2x+5

."

Question

Is the student's interpretation correct?

Accepted Answer

No, because if $\left(-2,5\right)$ , the equation is vertical and has no y-intercept

Answer

Yes, because the y-intercept is always $\left(-2,5\right)$

Answer

No, because the y-intercept is always

\frac{C}{A}

Answer

Yes, because setting

\frac{C}{A}

always gives

y=\frac{C}{B}

Question 29

Passage

A student states: “If two lines have the same slope but different y-intercepts, they intersect exactly once.”

Question

Is the student’s interpretation correct?

Accepted Answer

Yes, because different y-intercepts guarantee one intersection point

Answer

No, because those lines overlap and have infinitely many solutions

Answer

No, because same slope and different intercepts means the lines are parallel

Answer

Yes, because any two different lines must intersect

Question 30

Passage

A student says: “In $y=-4x+7$ , the line decreases because the y-intercept is negative."

Question

Is the student's interpretation correct?

Accepted Answer

No, because the line increases since the slope $3x$ is positive

Answer

Yes, because a negative y-intercept always makes a line decrease

Answer

Yes, because the sign of the y-intercept controls increasing vs. decreasing

Answer

No, because the slope is negative and the line must decrease

Question 31

Passage

A student claims: “In $y=-4x+7$ , the y-intercept is 10.”

Question

Is the student’s interpretation correct?

Accepted Answer

No, because isolating $3x$ gives $\left(-2,5\right)$ , so the y-intercept is 5

Answer

Yes, because the constant term is always the y-intercept even before solving for

\frac{C}{A}

Answer

Yes, because multiplying by 2 doubles the y-intercept

Answer

No, because the y-intercept is 6 since it’s the coefficient of

3x

Question 32

Passage

A student states: “To find an x-intercept, you should set $y=-4x+7$ .”

Question

Is the student’s interpretation correct?

Accepted Answer

No, because x-intercepts occur when $3x$

Answer

Yes, because setting $3x$ always gives the x-intercept value

Answer

No, because x-intercepts require setting $3x$

Answer

Yes, because intercepts are found by setting the same variable to 0

Question 33

Passage

A student says: “If a line has a positive x-intercept and a positive y-intercept, then its slope must be negative.”

Question

Is the student’s interpretation correct?

Accepted Answer

Yes, because going from a positive y-intercept to a positive x-intercept requires decreasing $3x$ as $\left(-2,5\right)$ increases

Answer

No, because two positive intercepts always create a positive slope

Answer

No, because intercept signs do not relate to slope direction at all

Answer

Yes, because two positive intercepts always create a vertical line

Question 34

Passage

A student explains: “Between (−1, 4) and (2, 10), y increases by 6 while x increases by 3, so the slope is 2.”

Question

Is the student’s interpretation correct?

Accepted Answer

Yes, because slope is $\left(-2,5\right)$

Answer

Yes, because any increase in $\left(-2,5\right)$ means slope must be 2

Answer

No, because the slope is 3 since $3x$

Answer

No, because slope should be $\left(-2,5\right)$

Question 35

Passage

A student says: “If the y-intercept is −3, then the line passes through (−3, 0).”

Question

Is the student’s interpretation correct?

Accepted Answer

No, because the y-intercept point is $\left(-2,5\right)$

Answer

Yes, because the y-intercept is always written as $\left(-2,5\right)$

Answer

Yes, because swapping coordinates does not change intercept meaning

Answer

No, because a y-intercept cannot be negative

Question 36

Passage

A student calculates the slope of a line as $\left(-2,5\right)$ . The student then concludes, “Because the slope is 2, the line must pass through the point $\left(4,-1\right)$ .”

Question

Is the student’s final conclusion logically valid?

Accepted Answer

No, because slope alone does not determine a unique point on the line

Answer

No, because a slope of 2 means the line must be horizontal

Answer

Yes, because any line with slope 2 must pass through the origin first

Answer

Yes, because a slope of 2 guarantees the x-intercept is 2

Question 37

Passage

A student is told that a line’s y-intercept is 3. The student concludes, “Therefore, the equation of the line must be $y=-4x+7$ .”

Question

Is the student’s final conclusion logically valid?

Accepted Answer

No, because many different lines can share the same y-intercept of 3

Answer

No, because a y-intercept of 3 implies the slope must be negative

Answer

Yes, because a y-intercept of 3 uniquely determines the equation $3x$

Answer

Yes, because the y-intercept always equals the slope in linear equations

Question 38

Passage

A line crosses the y-axis at $y=-4x+7$ and crosses the x-axis at

y-4=-2x+1\ \Rightarrow\ y=-2x+5

. A student concludes, “The slope of this line is

y=1

."

Question

Is the student's final conclusion correct?

Accepted Answer

No, because the slope is $\left(-2,5\right)$ , which is positive

Answer

No, because slopes cannot be determined from intercepts

Answer

Yes, because the slope equals

\frac{C}{A}

whenever one intercept is negative and the other is positive

Answer

Yes, because moving from $\left(-2,5\right)$ to $\left(4,-1\right)$ requires a negative slope

Question 39

Passage

A student says, “If a line has an x-intercept of 0, then it must have slope 0.”

Question

Is the student’s final conclusion logically valid?

Accepted Answer

No, because many non-horizontal lines pass through $3x$ and still have x-intercept 0

Answer

Yes, because an x-intercept of 0 implies the line is constant

Answer

No, because an x-intercept of 0 forces the slope to be undefined

Answer

Yes, because an x-intercept of 0 means the line lies on the x-axis

Question 40

Passage

A student concludes, “If a line has a negative slope, then its y-intercept must be negative as well.”

Question

Is the student’s final conclusion logically valid?

Accepted Answer

No, because the sign of the slope does not determine the sign of the y-intercept

Answer

Yes, because a decreasing line must cross the y-axis below the origin

Answer

No, because a negative slope implies the line is vertical

Answer

Yes, because slope and y-intercept always have the same sign

Question 41

Passage

Given the function $y=\frac{C}{B}$ , a student concludes, “The x-intercept is $-1$ because $-1$ is the constant term.”

Question

Is the student’s final conclusion correct?

Accepted Answer

No, because the x-intercept is found by setting $\left(-2,5\right)$ , which give $\left(4,-1\right)$

Answer

Yes, because x-intercepts are found by setting $3x$

Answer

Yes, because the constant term always equals the x-intercept

Answer

No, because the x-intercept must always be an integer

Question 42

Passage

A student looks at the equation $y=-4x+7$ and concludes, “This line has an undefined slope because it is a vertical line.”

Question

Is the student’s final conclusion correct?

Accepted Answer

No, because $\left(-2,5\right)$ is a horizontal line and horizontal lines have slope of 0

Answer

Yes, because $3x$ represents a vertical line

Answer

No, because $3x$ represents a vertical line

Answer

Yes, because slope does not exist when $3x$ equals $\left(-2,5\right)$

Question 43

Passage

A student computes the slope between $\left(-2,5\right)$ and $\left(4,-1\right)$ as $m=\frac{-1-5}{-2-4}=\frac{-6}{-6}=1$ , then concludes, "The slope is 0 because the denominator is 0."

Question

Is the student's final conclusion correct?

Accepted Answer

No, because division by 0 is undefined, indicating a vertical line

Answer

Yes, because the slope of any vertical line is 0

Answer

Yes, because a denominator of 0 makes the fraction equal 0

Answer

No, because the slope should be 7 whenever the y-values change by 7

Question 44

Passage

A student states, “If two lines intersect, then they must have different y-intercepts.”

Question

Is the student’s final conclusion logically valid?

Accepted Answer

No, because two lines can intersect at the y-axis and share the same y-intercept

Answer

Yes, because different y-intercepts are the only way lines can meet

Answer

Yes, because sharing a y-intercept would force the lines to be identical

Answer

No, because intersecting lines must have the same slope

Question 45

Passage

A student claims, “If two distinct lines have the same slope, then they never intersect.”

Question

Is the student’s final conclusion logically valid?

Accepted Answer

Yes, because distinct lines with the same slope are parallel and have no points in common

Answer

No, because lines with the same slope must share the same y-intercept

Answer

Yes, because equal slopes guarantee the lines overlap completely

Answer

No, because lines with the same slope always intersect once

Question 46

Passage

A student knows a line has slope $y=-4x+7$ . The student concludes, "If $y-4=-2x+1\ \Rightarrow\ y=-2x+5$ increases by $y=1$ , then $y$ increases by $2$ ."

Question

Is the student's final conclusion correct?

Accepted Answer

Yes, because $\left(-2,5\right)$ , so $\left(4,-1\right)$

Answer

Yes, because the y-intercept must be 2 for this to be true

Answer

No, because $3x$ must increase by 8 when $\left(-2,5\right)$ increases by 4

Answer

No, because a fractional slope means $3x$ decreases when $\left(-2,5\right)$ increases

Question 47

Passage

A student is told the y-intercept is 5 and concludes, “The line passes through $y=-4x+7$ .”

Question

Is the student’s final conclusion correct?

Accepted Answer

No, because a y-intercept of 5 corresponds to the point $3x$

Answer

Yes, because intercepts can be written in either coordinate order

Answer

Yes, because a y-intercept of 5 is written as $3x$ by convention

Answer

No, because a y-intercept does not correspond to a point on the graph

Question 48

Passage

A student sees $y=-4x+7$ and concludes, “This line has no y-intercept because the intercept is 0.”

Question

Is the student’s final conclusion correct?

Accepted Answer

No, because the line crosses the y-axis at $3x$ , so the y-intercept exists and equals 0

Answer

Yes, because an intercept of 0 means the graph does not cross that axis

Answer

Yes, because only positive intercepts count as intercepts

Answer

No, because a y-intercept of 0 makes the line vertical

Question 49

Passage

A student concludes, “If the slope of a linear function is 0, then the function is constant and its graph is horizontal.”

Question

Is the student’s final conclusion correct?

Accepted Answer

Yes, because slope 0 means y does not change as x changes

Answer

No, because a constant function must have undefined slope

Answer

Yes, because slope 0 means the line passes through the origin

Answer

No, because slope 0 corresponds to a vertical line

Question 50

Passage

A student claims, “Knowing one point on a line and the slope determines exactly one unique line.”

Question

Is the student’s final conclusion correct?

Accepted Answer

Yes, because there is exactly one line through a given point with a specified slope

Answer

Yes, because slope alone determines a unique line regardless of the point

Answer

No, because slope and one point can describe multiple different lines

Answer

No, because a second point is always required to determine a unique line

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