Slope of a Line

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Slope of a Line

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Question 1 of 9

Find the gradient of the line between the points

A (1, 4)

$A(1,4)$ and

B (3, 2)

$B(3,2)$

$m =$ $m=$ (-1)

Incorrect

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The gradient $(m)$ $(m)$ refers to the “slope” or “slant” of a line

Method One

Gradient Formula

m = \frac{r i s e}{r u n}

$m=\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

To find the gradient

(m)

$(m)$ , measure the rise and run between two points on the line

From point

(3, 2)

$(3, 2)$ to

(1, 4)

$(1, 4)$ the rise is $2$ $2$ units and the run is $- 2$ $-2$ units

The run is negative as the run goes to the left

$m$ $m$	$=$ $=$	$\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

	$=$ $=$	$\frac{\color{#9a00c7}{2}}{\color{#00880a}{-2}}$

$m$ $m$	$=$ $=$	$- 1$ $-1$

m = - 1

$m=-1$

Method Two

Gradient Formula

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

$m=\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$

First, choose any two coordinates on the line to use in the Gradient Formula

We have chosen the points:

(1, 4)

$(1,4)$ and

(3, 2)

$(3,2)$

Label the coordinates we chose from above

(

$($ $1$ $1$

,

$,$ $4$ $4$

) = (

$) = ($ $x_{1}$ $x_1$

,

$,$ $y_{1}$ $y_1$

)

$)$

(

$($ $3$ $3$

,

$,$ $2$ $2$

) = (

$) = ($ $x_{2}$ $x_2$

,

$,$ $y_{2}$ $y_2$

)

$)$

Solve for the gradient

(m)

$(m)$ by substituting the two coordinates into the formula

$m$ $m$	$=$ $=$	$\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$	Gradient Formula

	$=$ $=$	$\frac{\color{#9a00c7}{2}-\color{#9a00c7}{4}}{\color{#00880a}{3}-\color{#00880a}{1}}$	Plug in the coordinates

	$=$ $=$	$\frac{- 2}{2}$ $(-2)/(2)$	Evaluate

$m$ $m$	$=$ $=$	$- 1$ $-1$	$- 2 \div 2 = - 1$ $-2 divide 2=-1$

m = - 1

$m=-1$

Question 2 of 9

Find the gradient of the line shown below

$m =$ $m=$ (3/2, 1.5)

Incorrect

Help Video

The gradient $(m)$ $(m)$ refers to the “slope” or “slant” of a line

Method One

Gradient Formula

m = \frac{r i s e}{r u n}

$m=\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

To find the gradient

(m)

$(m)$ , measure the rise and run between two points on the line

From point

(2, 2)

$(2, 2)$ to

(4, 5)

$(4, 5)$ the rise is $3$ $3$ units and the run is $2$ $2$ units

$m$ $m$	$=$ $=$	$\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

	$=$ $=$	$\frac{\color{#9a00c7}{3}}{\color{#00880a}{2}}$

$m$	$=$	$3/2$

$m=3/2$

Method Two

Gradient Formula

$m=\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$

First, choose any two coordinates on the line to use in the Gradient Formula

We have chosen the points:

$(2,2)$ and

$(4,5)$

Label the coordinates we chose from above

$($ $2$

$,$ $2$

$) = ($ $x_1$

$,$ $y_1$

$)$

$($ $4$

$,$ $5$

$) = ($ $x_2$

$,$ $y_2$

$)$

Solve for the gradient

$(m)$ by substituting the two coordinates into the formula

$m$	$=$	$\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$	Gradient Formula

	$=$	$\frac{\color{#9a00c7}{5}-\color{#9a00c7}{2}}{\color{#00880a}{4}-\color{#00880a}{2}}$	Plug in the coordinates

	$=$	$3/2$	Evaluate

$m=3/2$

Question 3 of 9

Find the gradient of the line shown below

$m=$ (2)

Incorrect

Help Video

The gradient $(m)$ refers to the “slope” or “slant” of a line

Method One

Gradient Formula

$m=\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

To find the gradient

$(m)$ , measure the rise and run between two points on the line

From point

$(-1, 0)$ to

$(0, 2)$ the rise is $2$ units and the run is $1$ unit

$m$	$=$	$\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

	$=$	$\frac{\color{#9a00c7}{2}}{\color{#00880a}{1}}$

$m$	$=$	$2$

$m=2$

Method Two

Gradient Formula

$m=\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$

First, choose any two coordinates on the line to use in the Gradient Formula

We have chosen the points:

$(0,2)$ and

$(-1,0)$

Label the coordinates we chose from above

$($ $0$

$,$ $2$

$) = ($ $x_1$

$,$ $y_1$

$)$

$($ $-1$

$,$ $0$

$) = ($ $x_2$

$,$ $y_2$

$)$

Solve for the gradient

$(m)$ by substituting the two coordinates into the formula

$m$	$=$	$\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$	Gradient Formula

	$=$	$\frac{\color{#9a00c7}{0}-\color{#9a00c7}{2}}{\color{#00880a}{-1}-\color{#00880a}{0}}$	Plug in the coordinates

	$=$	$(-2)/(-1)$	Evaluate
$m$	$=$	$2$	$-2 divide -1=2$

$m=2$

Question 4 of 9

4. Question
Find the gradient of the line shown below
- $m=$ (2)
Hint

Help Video

Correct

Correct!

Incorrect

Help Video

The gradient $(m)$ refers to the “slope” or “slant” of a line

Gradient Formula

$m=\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

To find the gradient $(m)$ , measure the rise and run between two points on the line

From point $(0, 0)$ to $(1, 2)$ the rise is $2$ units and the run is $1$ unit

$m$ $=$ $\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

$=$ $\frac{\color{#9a00c7}{2}}{\color{#00880a}{1}}$

$m$ $=$ $2$

$m=2$
Question 5 of 9

5. Question
Find the gradient of the line shown below
- $m=$ (1)
Hint

Help Video

Correct

Excellent!

Incorrect

Help Video

The gradient $(m)$ refers to the “slope” or “slant” of a line

Gradient Formula

$m=\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

To find the gradient $(m)$ , measure the rise and run between two points on the line

From point $(0, 0)$ to $(2, 2)$ the rise is $2$ units and the run is $2$ units

$m$ $=$ $\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

$=$ $\frac{\color{#9a00c7}{2}}{\color{#00880a}{2}}$

$m$ $=$ $1$

$m=1$
Question 6 of 9

6. Question
Find the gradient of the line shown below
- $m=$ (1/4, 0.25)
Hint

Help Video

Correct

Keep Going!

Incorrect

Help Video

The gradient $(m)$ refers to the “slope” or “slant” of a line

Gradient Formula

$m=\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

To find the gradient $(m)$ , measure the rise and run between two points on the line

From point $(0, 0)$ to $(4, 1)$ the rise is $1$ unit and the run is $4$ units

$m$ $=$ $\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

$=$ $\frac{\color{#9a00c7}{1}}{\color{#00880a}{4}}$

$m$ $=$ $1/4$

$m=1/4$
Question 7 of 9

7. Question
Find the gradient of the line shown below
- $m=$ (-1/2, -0.5)
Hint

Help Video

Correct

Fantastic!

Incorrect

Help Video

The gradient $(m)$ refers to the “slope” or “slant” of a line

Gradient Formula

$m=\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

To find the gradient $(m)$ , measure the rise and run between two points on the line

From point $(4, 0)$ to $(0, 2)$ the rise is $2$ units and the run is $-4$ units

$m$ $=$ $\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

$=$ $\frac{\color{#9a00c7}{2}}{\color{#00880a}{-4}}$

$=$ $-2/4$

$m$ $=$ $-1/2$ Reduce to lowest terms

$m=-1/2$

Question 8 of 9

Is the shape below a parallelogram?

1. No
2. Yes

Incorrect

Help Video

A parallelogram is a quadrilateral with opposite sides parallel. Parallel lines have the same gradients.

Gradient Formula

$m=\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$

First, label the given coordinates

$A($ $1$

$,$ $2$

$) = ($ $x_1$

$,$ $y_1$

$)$

$B($ $5$

$,$ $2$

$) = ($ $x_1$

$,$ $y_1$

$)$

$C($ $3$

$,$ $-2$

$) = ($ $x_2$

$,$ $y_2$

$)$

$D($ $-1$

$,$ $-2$

$) = ($ $x_2$

$,$ $y_2$

$)$

Solve for the gradient

$(m)$ by substituting the pairs of coordinates for lines

$AD$ and

$BC$ into the formula

$m_{AD}$	$=$	$\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$

	$=$	$\frac{\color{#9a00c7}{-2}-\color{#9a00c7}{2}}{\color{#00880a}{-1}-\color{#00880a}{1}}$

	$=$	$\frac{\color{#9a00c7}{-4}}{\color{#00880a}{-2}}$

$m_{AD}$	$=$	$2$

$m_{BC}$	$=$	$\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$

	$=$	$\frac{\color{#9a00c7}{-2}-\color{#9a00c7}{2}}{\color{#00880a}{3}-\color{#00880a}{5}}$

	$=$	$\frac{\color{#9a00c7}{-4}}{\color{#00880a}{-2}}$

$m_{BC}$	$=$	$2$

Identify whether the quadrilateral is a parallelogram.

$m_(AD) = m_(BC)$ so lines

$AD$ and

$BC$ are parallel, and therefore the given quadrilateral is a parallelogram.

The quadrilateral is a parallelogram.

Question 9 of 9

9. Question
Are the two lines parallel?
- 1. No
- 2. Yes
Hint

Help Video

Correct

Nice Job!

Incorrect

Help Video

Parallel lines have the same gradients.

Gradient Formula

$m=\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

To find the gradient $(m)$ , measure the rise and run between two points on the line

Black line: From point $(4, 0)$ to $(0, 4)$ the rise is $4$ units and the run is $-4$ units

$m$ $=$ $\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

$=$ $\frac{\color{#9a00c7}{4}}{\color{#00880a}{-4}}$

$m$ $=$ $-1$

Red line: From point $(-2, 0)$ to $(0, -2)$ the rise is $-2$ units and the run is $2$ units

$m$ $=$ $\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

$=$ $\frac{\color{#9a00c7}{-2}}{\color{#00880a}{2}}$

$m$ $=$ $-1$

The two gradients are equal, so the lines are parallel.

The lines are parallel.

$m$	$=$	$\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

	$=$	$\frac{\color{#9a00c7}{1}}{\color{#00880a}{4}}$

$m$	$=$	$1/4$

$m$	$=$	$\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

	$=$	$\frac{\color{#9a00c7}{2}}{\color{#00880a}{-4}}$

	$=$	$-2/4$

$m$	$=$	$-1/2$	Reduce to lowest terms

$m$	$=$	$\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

	$=$	$\frac{\color{#9a00c7}{4}}{\color{#00880a}{-4}}$

$m$	$=$	$-1$

$m$	$=$	$\frac{\color{#9a00c7}{rise}}{\color{#00880a}{run}}$

	$=$	$\frac{\color{#9a00c7}{-2}}{\color{#00880a}{2}}$

$m$	$=$	$-1$

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Quiz summary

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1. Question

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Well Done!

Gradient Formula

Gradient Formula

2. Question

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Great Work!

Gradient Formula

Gradient Formula

3. Question

Hint

Nice Job!

Gradient Formula

Gradient Formula

4. Question

Hint

Correct!

Gradient Formula

5. Question

Hint

Excellent!

Gradient Formula

6. Question

Hint

Keep Going!

Gradient Formula

7. Question

Hint

Fantastic!

Gradient Formula

8. Question

Hint

Excellent!

Gradient Formula

9. Question

Hint

Nice Job!

Gradient Formula

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