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Slope: Core practice

10 practice problems for this lesson. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

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Problem 1 of 10

Problem 1 Grid measurements

Find the slope of the line in the figure.

A line through two marked points.

Text description of this figure

A coordinate grid drawn with the same length for one unit on both axes. The x-axis runs from negative 4 to 6, with vertical gridlines and numbered ticks at negative 4, negative 2, 0, 2, 4 and 6, so the vertical gridlines are 2 units apart. The y-axis runs from negative 3 to 9, with horizontal gridlines and numbered ticks at negative 3, 0, 3, 6 and 9, so the horizontal gridlines are 3 units apart. The origin is labeled 0. A straight line with arrowheads at both ends falls from left to right across the grid, from the top left corner of the grid at negative 4, 9, through the y-axis at 3, down to the bottom edge at 4, negative 3. Two points on the line are marked with dots and carry no labels: one where the gridlines x equals negative 2 and y equals 6 meet, the point (negative 2, 6), and one on the x-axis at the point (2, 0). No slope triangle or slope label is drawn.
1. Hint 1
  Use coordinate changes, not just the number of drawn grid spaces.
2. Hint 2
  Read the coordinates of the two marked points and divide the signed rise by the signed run.
Answer

$m=-\frac32$ .

Full solution

The marked points are $(-2,6)$ and $(2,0)$ .
Moving from the first to the second gives a rise of $-6$ and a run of $4$ :
$m=\frac{-6}{4}$ $m=-\frac32$
The line falls to the right, agreeing with the negative sign.

Answer

$m=-\frac32$ .

Key idea

Slope compares changes measured in coordinate units.
Problem 1: Mark complete
Problem 2 Negative coordinates

Find the slope of the line through $(-7,4)$ and $(5,-5)$ .
1. Hint 1
  Slope is the change in $y$ divided by the change in $x$ , both taken in the same order.
2. Hint 2
  With $(-7,4)$ as the first point, the rise is $-5-4$ and the run is $5-(-7)$ .
Answer

$m=-\frac34$ , or $-0.75$ .

Full solution

Take $(-7,4)$ as the first point and $(5,-5)$ as the second.
The rise is $-5-4$ , which is $-9$ .
The run is $5-(-7)$ , which is $12$ .

So the slope is
$m=\frac{-9}{12}$ $m=-\frac34$
Moving right from $(-7,4)$ to $(5,-5)$ , the height drops, so the line falls to the right, which agrees with the negative sign.

Answer

$m=-\frac34$ , or $-0.75$ .

Key idea

Subtracting a negative coordinate adds its size, so write each subtraction out in full before simplifying.
Problem 2: Mark complete
Problem 3 Point restrictions

Classify the slope of each line described by its points: A has second coordinate $7$ for every first coordinate; B has first coordinate $-4$ for every second coordinate; C has $y$ decrease whenever $x$ increases.
1. Hint 1
  Slope depends on whether the rise or the run is zero, and on the signs of nonzero changes.
2. Hint 2
  A fixed second coordinate gives no rise; a fixed first coordinate gives no run.
Answer

A: zero; B: undefined; C: negative.

Full solution

For A, two distinct points have zero rise and nonzero run, so
$m_A=0$
For B, distinct points have zero run, and division by zero has no value.

For C, a move to the right has positive run and negative rise, so the ratio is negative.
These descriptions respectively give a horizontal line, a vertical line, and a line falling to the right.

Answer

A: zero; B: undefined; C: negative.

Key idea

A fixed height gives zero slope, a fixed column gives undefined slope, and a fall to the right gives negative slope.
Problem 3: Mark complete
Problem 4 Water readings

Water drains from a tank. Its height is 12 cm at 0 minutes, 9 cm at 3 minutes and 5 cm at 7 minutes. Do the three readings lie on one straight-line graph? If they do, and the height keeps changing at the same rate, find the height at 10 minutes.
1. Hint 1
  Compare the height change per minute in each interval.
2. Hint 2
  If the two rates agree, continue at that rate for the three minutes after the last reading.
Answer

Yes; the height at 10 minutes is 2 cm.

Full solution

The first interval gives
$m_1=\frac{9-12}{3-0}$ $m_1=-1$
The second gives
$m_2=\frac{5-9}{7-3}$ $m_2=-1$
The two intervals have the same slope and share the reading at 3 minutes, so the three readings lie on one straight line.

Three more minutes lower the height by 3 cm, so it reaches $5-3=2$ cm.
From the starting reading, $12-10=2$ cm gives an independent check.

Answer

Yes; the height at 10 minutes is 2 cm.

Key idea

Equal slopes over unequal time intervals support one constant rate of change.
Problem 4: Mark complete
Problem 5 Route drawing

The figure joins $A$ to $B$ and $B$ to $C$ . Find the slope of each segment and decide whether the route follows one straight line.

The route from $A$ to $B$ to $C$ .

Text description of this figure

A coordinate grid with the same length for one unit on both axes and a gridline at every whole number. The x-axis runs from negative 5 to 4 and the y-axis from negative 3 to 4, both with arrowheads at each end, a numbered tick at every whole number, and the origin labeled 0. Three points are marked with dots and labeled with their letters only: A at (negative 4, negative 3), B at the origin (0, 0), and C at (3, 2). A segment joins A to B and a second segment joins B to C; neither is extended past its endpoints. No angles, slope triangles or coordinates are shown.
1. Hint 1
  Each part of one nonvertical straight line must have the same slope.
2. Hint 2
  Read the coordinates and keep the subtraction order consistent on each segment.
Answer

$m_{AB}=\frac34$ and $m_{BC}=\frac23$ ; the route does not follow one straight line.

Full solution

The points are $A(-4,-3)$ , $B(0,0)$ , and $C(3,2)$ .
Their slopes are
$m_{AB}=\frac{0-(-3)}{0-(-4)}$ $m_{AB}=\frac34$ $m_{BC}=\frac{2-0}{3-0}$ $m_{BC}=\frac23$
The unequal slopes show a change in direction at $B$ , so the route is not one straight line.

Answer

$m_{AB}=\frac34$ and $m_{BC}=\frac23$ ; the route does not follow one straight line.

Key idea

Different slopes on adjoining segments reveal a bend in a route.
Problem 5: Mark complete
Problem 6 Ramp measurements

A straight ramp rises 45 cm over a horizontal distance of 3 meters. Find its slope, then find its rise over the first 1.2 meters of horizontal distance.
1. Hint 1
  Convert the two measurements to the same unit before taking their ratio.
2. Hint 2
  The same rise-to-run ratio holds over a shorter part of the straight ramp.
Answer

Slope $0.15$ (or $\frac{3}{20}$ ); rise $0.18$ meters (18 cm).

Full solution

In meters, the full rise is $0.45$ and the run is $3$ , so
$m=\frac{0.45}{3}$ $m=0.15$
Measuring both in centimeters gives $\frac{45}{300}$ , the same slope.

The shorter rise is
$0.15(1.2)=0.18$
It is $0.18$ meters, or 18 cm.
The shorter run is $0.4$ of the full run, and $0.4(0.45)=0.18$ checks the rise.

Answer

Slope $0.15$ (or $\frac{3}{20}$ ); rise $0.18$ meters (18 cm).

Key idea

Slope requires consistent units and gives proportional rises along a straight ramp.
Problem 6: Mark complete
Problem 7 Grid locations

The figure shows a line through two marked points. List every point on the line inside the displayed grid whose two coordinates are integers. Explain how the slope helps you make a complete list.

A line through two marked points on a unit grid.

Text description of this figure

A coordinate grid with the same length for one unit on both axes and a gridline at every whole number. The x-axis runs from negative 5 to 6 and the y-axis from negative 4 to 4, both with arrowheads at each end, a numbered tick at every whole number, and the origin labeled 0. A straight line falls from left to right across the whole grid and stops at its edges, with an arrowhead at each end: it enters at the left edge, x equals negative 5, between y equals 3 and 4, and leaves at the right edge, x equals 6, between y equals negative 3 and negative 4. Two points on the line are marked with dots and carry no letters or coordinates: (negative 4, 3) and (2, negative 1). No other points are marked.
1. Hint 1
  Find the slope from the two marked points; every step with that rise-to-run ratio stays on the line.
2. Hint 2
  Find the smallest whole-number run that gives a whole-number rise, then step both ways to the grid edges.
Answer

$(-4,3)$ , $(-1,1)$ , $(2,-1)$ , and $(5,-3)$ ; they are spaced by runs of 3 and rises of $-2$ , the smallest whole-number step for slope $-\frac23$ .

Full solution

The marked points are $(-4,3)$ and $(2,-1)$ , giving
$m=\frac{-1-3}{2-(-4)}$ $m=-\frac23$
Each step of 3 right and 2 down reaches another point with integer coordinates.

Starting at $(-4,3)$ gives $(-1,1)$ , $(2,-1)$ , and $(5,-3)$ .
A step left leaves the grid, as does the next step right.
A run of 1 or 2 produces a noninteger rise, so there are no skipped integer-coordinate points.

Answer

$(-4,3)$ , $(-1,1)$ , $(2,-1)$ , and $(5,-3)$ ; they are spaced by runs of 3 and rises of $-2$ , the smallest whole-number step for slope $-\frac23$ .

Key idea

On a line through at least one integer-coordinate point, a reduced slope $\frac{p}{q}$ with $q$ positive spaces those points by runs of $q$ and rises of $p$ .
Problem 7: Mark complete
Problem 8 Shifted coordinate records

For every point on a nonvertical line, a recorder adds 5 to the first coordinate and subtracts 2 from the second. A student claims this changes the slope. Is the claim correct? Justify your answer for any two distinct original points.
1. Hint 1
  Slope uses coordinate differences rather than the individual coordinates.
2. Hint 2
  Compare what happens when the same adjustment appears in both terms of a difference.
Answer

No; the slope is unchanged.

Full solution

Let the original points be $(a,b)$ and $(c,d)$ with $a\ne c$ .
The new rise is
$(d-2)-(b-2)=d-b$
The new run is
$(c+5)-(a+5)=c-a$

Both changes equal their original values.
The run is nonzero because the original line is nonvertical and the points are distinct.
Thus both slopes are $(d-b)/(c-a)$ , and the recorded line has the same slope.

Answer

No; the slope is unchanged.

Key idea

Adding fixed amounts to coordinates preserves the differences that determine slope.
Problem 8: Mark complete
Problem 9 Two distinct points

A student says, "If a slope calculation has no value, the two points do not determine a line." Test this claim using two distinct points that you choose, and explain what happens to the slope calculation.
1. Hint 1
  Which kind of line makes the slope formula fail?
2. Hint 2
  Two different points can share an $x$ -coordinate while lying at different heights.
Answer

False; for example, $(2,1)$ and $(2,4)$ determine $x=2$ , whose slope is undefined.

Full solution

Take $(2,1)$ and $(2,4)$ .
Their slope calculation is
$\frac{4-1}{2-2}=\frac30$
Division by zero has no value.
Nevertheless, both points lie on the vertical line $x=2$ , which they determine uniquely.

The missing slope number reflects a zero run, not the absence of a line.

Answer

False; for example, $(2,1)$ and $(2,4)$ determine $x=2$ , whose slope is undefined.

Key idea

A vertical line exists even though its rise cannot be divided by its zero run.
Problem 9: Mark complete
Problem 10 Joined movements

A path is made of two straight moves, one after the other, both going right. Their runs are $r$ and $s$ , both positive, and each move has slope $m$ . Does the move from the start of the path to its end also have slope $m$ , even when the runs differ? Justify your answer using $r$ , $s$ and $m$ .
1. Hint 1
  Turn each slope statement into a rise using its own run.
2. Hint 2
  Add the rises and add the runs before forming the combined ratio.
Answer

Yes; the combined slope is $m$ .

Full solution

The rises are $mr$ and $ms$ , while the combined run is $r+s$ .
Since both runs are positive, $r+s$ is nonzero and division is valid.
The combined slope is
$\frac{mr+ms}{r+s}$
Factoring the numerator gives
$\frac{m(r+s)}{r+s}=m$
Thus unequal runs still produce the same slope.

This also applies to a negative or zero $m$ , since the rises carry its sign.

Answer

Yes; the combined slope is $m$ .

Key idea

Adding moves with the same slope preserves that slope when the combined run is nonzero.
Problem 10: Mark complete

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