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What is the slope of the line on the geoboard shown?
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 5 and the peg in column 5, row 2, forming a line.

Solution

Use the definition of slope: m = rise run .

Start at the left peg and count the spaces up and to the right to reach the second peg.
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 5 and the peg in column 5, row 2, forming a right triangle. The 1, 2 peg forms the vertex of the 90 degree angle and the line from the 1, 5 peg to the 5, 2 peg forms the hypotenuse of the triangle. The line from the 1, 2 peg to the 1, 5 peg is labeled “3”. The line from the 1, 2 peg to the 5, 2 peg is labeled “4”.

The rise is 3. m = 3 run The run is 4. m = 3 4 The slope is 3 4 .

This means that the line rises 3 units for every 4 units of run.

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What is the slope of the line on the geoboard shown?

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 5 and the peg in column 4, row 1, forming a line.

4 3

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What is the slope of the line on the geoboard shown?

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 5, row 3, forming a line.

1 4

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What is the slope of the line on the geoboard shown?
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 4, row 4, forming a line.

Solution

Use the definition of slope: m = rise run .

Start at the left peg and count the units down and to the right to reach the second peg.
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3, the peg in column 1, row 4 and the peg in column 4, row 4, forming a right triangle. The 1, 3 peg forms the vertex of the 90 degree angle and the line from the 1, 4 peg to the 4, 4 peg forms the hypotenuse of the triangle. The line from the 1, 3 peg to the 1, 4 peg is labeled “negative 1”. The line from the 1, 4 peg to the 4, 4 peg is labeled “3”.

The rise is −1 . m = −1 run The run is 3. m = −1 3 m = 1 3 The slope is 1 3 .

This means that the line drops 1 unit for every 3 units of run.

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What is the slope of the line on the geoboard?

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2 and the peg in column 4, row 4, forming a line.

2 3

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What is the slope of the line on the geoboard?

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1 and the peg in column 4, row 5, forming a line.

4 3

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Notice that in [link] the slope is positive and in [link] the slope is negative. Do you notice any difference in the two lines shown in [link] (a) and [link] (b)?

The figure shows two grids of evenly spaced pegs, one labeled (a) and one labeled (b). There are 5 columns and 5 rows of pegs in each grid. In the (a) grid, a rubber band is stretched between the peg in column 1, row 5 and the peg in column 5, row 2, forming a line. Below this grid is the slope of a line defined as m equals 3 fourths. In the (b) grid, a rubber band is stretched between the peg in column 1, row 3 and the peg in column 4, row 4, forming a line. Below this grid is the slope of a line defined as m equals negative 1 third.

We ‘read’ a line from left to right just like we read words in English. As you read from left to right, the line in [link] (a) is going up; it has positive slope    . The line in [link] (b) is going down; it has negative slope    .

Positive and negative slopes

The figure shows two lines side-by-side. The line on the left is a diagonal line that rises from left to right. It is labeled “Positive slope”. The line on the right is a diagonal line that drops from left to right. It is labeled “Negative slope”.

Use a geoboard to model a line with slope 1 2 .

Solution

To model a line on a geoboard, we need the rise and the run.

Use the slope formula. m = rise run Replace m with 1 2 . 1 2 = rise run

So, the rise is 1 and the run is 2.

Start at a peg in the lower left of the geoboard.

Stretch the rubber band up 1 unit, and then right 2 units.
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3, the peg in column 1, row 4 and the peg in column 3, row 3, forming a right triangle. The 1, 3 peg forms the vertex of the 90 degree angle and the line from the 1, 4 peg to the 3, 3 peg forms the hypotenuse of the triangle. The line from the 1, 3 peg to the 1, 4 peg is labeled “1”. The line from the 1, 3 peg to the 3, 3 peg is labeled “2”.

The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is 1 2 .

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Model the slope m = 1 3 . Draw a picture to show your results.

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 3, the peg in column 2, row 4 and the peg in column 5, row 3, forming a right triangle. The 2, 3 peg forms the vertex of the 90 degree angle and the line from the 2, 4 peg to the 5, 3 peg forms the hypotenuse of the triangle.

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Model the slope m = 3 2 . Draw a picture to show your results.

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1, the peg in column 1, row 4 and the peg in column 3, row 1, forming a right triangle. The 1, 1 peg forms the vertex of the 90 degree angle and the line from the 1, 4 peg to the 3, 1 peg forms the hypotenuse of the triangle.

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Use a geoboard to model a line with slope −1 4 .

Solution

Use the slope formula. m = rise run Replace m with −1 4 . −1 4 = rise run

So, the rise is −1 and the run is 4.

Since the rise is negative, we choose a starting peg on the upper left that will give us room to count down.

We stretch the rubber band down 1 unit, then go to the right 4 units, as shown.
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 3 and the peg in column 5, row 3, forming a right triangle. The 1, 3 peg forms the vertex of the 90 degree angle and the line from the 1, 2 peg to the 5, 3 peg forms the hypotenuse of the triangle. The line from the 1, 2 peg to the 1, 3 peg is labeled “negative 1”. The line from the 1, 3 peg to the 5, 3 peg is labeled “4”.

The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is −1 4 .

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Model the slope m = −2 3 . Draw a picture to show your results.

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 3, the peg in column 2, row 5 and the peg in column 3, row 5, forming a right triangle. The 2, 5 peg forms the vertex of the 90 degree angle and the line from the 2, 3 peg to the 3, 5 peg forms the hypotenuse of the triangle.

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Model the slope m = −1 3 . Draw a picture to show your results.

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1, the peg in column 1, row 2 and the peg in column 4, row 2, forming a right triangle. The 1, 2 peg forms the vertex of the 90 degree angle and the line from the 1, 1 peg to the 4, 2 peg forms the hypotenuse of the triangle.

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Use m = rise run To find the slope of a line from its graph

Now, we’ll look at some graphs on the x y -coordinate plane and see how to find their slopes. The method will be very similar to what we just modeled on our geoboards.

To find the slope, we must count out the rise and the run. But where do we start?

We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.

How to use m = rise run To find the slope of a line from its graph

Find the slope of the line shown.

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 6 and the y-axis runs from negative 4 to 2. A line passes through the points (0, negative 3) and (5, 1).

Solution

This table has three columns and four rows. The first row says, “Step 1. Locate two points on the graph whose coordinates are integers. Mark (0, negative 3) and (5, 1).” To the right is a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 6. The y-axis of the plane runs from negative 4 to 2. The points (0, negative 3) and  (5, 1) are plotted. The second row says, “Step 2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point. Starting at (0, negative 3), sketch a right triangle to (5, 1).” In the graph on the right, an additional point is plotted at (0, 1). The three points form a right triangle, with the line from (0, negative 3) to (5, 1) forming the hypotenuse and the lines from (0, negative 3) to (0, 1) and (0, 1) to (5, 1) forming the legs. The third row then says, “Step 3. Count the rise and the run on the legs of the triangle.” The rise is 4 and the run is 5. The fourth row says, “Step 4. Take the ratio of the rise to run to find the slope. Use the slope formula. Substitute the values of the rise and run.” To the right is the slope formula, m equals rise divided by run. The slope of the line is 4 divided by 5, or four fifths. This means that y increases 4 units as x increases 5 units.
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Find the slope of the line shown.

The graph shows the x y coordinate plane. The x-axis runs from negative 8 to 1 and the y-axis runs from negative 1 to 4. A line passes through the points (negative 5, 1) and (0, 3).

2 5

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Find the slope of the line shown.

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 4. A line passes through the points (0, negative 1) and (4, 2).

3 4

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Find the slope of a line from its graph using m = rise run .

  1. Locate two points on the line whose coordinates are integers.
  2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
  3. Count the rise and the run on the legs of the triangle.
  4. Take the ratio of rise to run to find the slope, m = rise run .

Questions & Answers

Aziza is solving this equation-2(1+x)=4x+10
Sechabe Reply
No. 3^32 -1 has exactly two divisors greater than 75 and less than 85 what is their product?
KAJAL Reply
x^2+7x-19=0 has Two solutions A and B give your answer to 3 decimal places
Adedamola Reply
please the answer to the example exercise
Patricia Reply
3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
Edwin Reply
.473
Angelita
?
Angelita
John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18.
DaYoungan Reply
p-2/3=5/6 how do I solve it with explanation pls
Adedamola Reply
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
don't understand answer
Cindy
4.5
Ruth
is y=7/5 a solution of 5y+3=10y-4
Adedamola Reply
yes
James
don't understand answer
Cindy
Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have?
Rhonda Reply
Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
Gabriel Reply
is that a negative 4 or positive 4?
Felix
y = mx + b
Felix
if negative -4, then -4=8/9(0) + b
Felix
-4=b
Felix
if positive 4, then 4=b
Felix
then plug in y=8/9x - 4 or y=8/9x+4
Felix
Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make?
Cherry Reply
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up
Makaila Reply
$1750
Cindy
the sum of two Numbers is 19 and their difference is 15
Abdulai Reply
2, 17
Jose
interesting
saw
4,2
Cindy
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Rafi Reply
hola saben como aser un valor de la expresión
NAILEA
integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
6
Cindy
Practice Key Terms 7

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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