Question

4.1 & 4.2: Write the equation of a line in slope intercept form given the graph.

Answer 1

the equations for the two lines:

1. First line:
`\( y = -x + 2 \)`

2. Second line:
`\( y = 3x \)`

To write the equation of a line in slope-intercept form
`\( y = mx + b \)`, we need two things:

1. The slope
`(\( m \))` of the line, which can be calculated using two points on the line with the formula
`\( m = \frac{\text{rise}}{\text{run}} = (y_2 - y_1)/(x_2 - x_1) \).`

2. The y-intercept
`(\( b \))`, which is the value of
`\( y \)` where the line crosses the y-axis
`(\( x = 0 \))`.

For the first image, using the points (2,0) and (0,2):

1. Calculate the slope
`(\( m \))`:

`\[ m = (2 - 0)/(0 - 2) = (2)/(-2) = -1 \]`

2. Since the line crosses the y-axis at (0,2), the y-intercept
`(\( b \))` is 2.

Therefore, the equation of the line for the first image is:

`\[ y = -x + 2 \]`

For the second image, using the points
`(1,3)` and
`(-1,-3)`:

1. Calculate the slope
`(\( m \))`:

`\[ m = (-3 - 3)/(-1 - 1) = (-6)/(-2) = 3 \]`

2. To find the y-intercept
`(\( b \))`, we can use the point (1,3) and the slope we just calculated. Plugging the point into the slope-intercept form gives us:

`\[ 3 = 3(1) + b \]`

`\[ b = 3 - 3 \]`

`\[ b = 0 \]`

Therefore, the equation of the line for the second image is:

`\[ y = 3x \]`

So we have the equations for the two lines:

1. First line:
`\( y = -x + 2 \)`

2. Second line:
`\( y = 3x \)`

Answer 2

We have the following:

1.

the formula for the slope is:

`m=(y_2-y_1)/(x_2-x_1)`

The points are (-5, 4) and (5, - 2)

replacing:

`m=(-2-4)/(5-(-5))=(-6)/(5+5)=-(6)/(10)=-(3)/(5)`

The y-intercept (b) is 1, the equation of a line in slope intercept form is:

`y=mx+b`

replacing:

`y=-(3)/(5)x+1`

2.

The points are (- 1, - 3) and (1, 3)

replacing

`m=(3-(-3))/(1-(-1))=(3+3)/(1+1)=(6)/(2)=3`

The y-intercept (b) is 0, replacing:

`y=3x`