Question

6.1 slope intercept form given slope/y-intercept or graph

1. slope intercept form:
2. write an equation in slope intercept form of a line with slope = 4 and y-intercept: (0, −3)
3. write an equation in slope intercept form of the line graphed below.

6.2 slope intercept form given point/slope
write an equation in slope intercept form using the given information.

1. slope = −2 through point (−1,5)
2. ( m = \frac{1}{3} ) through point (6, −2)

6.3 slope intercept form given two points
write an equation in slope intercept form using the given information.

1. through (3, 4) and (5, 8)
2. through (4, 5) and (8, 3)

Explanation:

Problem 2:

Step1: Recall slope - intercept form

The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.
We are given that the slope $m = 4$ and the y - intercept is at the point $(0,-3)$, so $b=-3$.

Step2: Substitute $m$ and $b$ into the formula

Substitute $m = 4$ and $b=-3$ into $y=mx + b$. We get $y=4x-3$.

Step1: Identify two points on the line

From the graph, we can see that the line passes through the points $(0,2)$ (the y - intercept) and $(1,0)$.

Step2: Calculate the slope ($m$)

The formula for slope between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Using the points $(0,2)$ and $(1,0)$, we have $x_1 = 0,y_1 = 2,x_2=1,y_2 = 0$. So $m=\frac{0 - 2}{1-0}=\frac{-2}{1}=-2$.

Step3: Identify the y - intercept ($b$)

The y - intercept is the value of $y$ when $x = 0$. From the point $(0,2)$, we know that $b = 2$.

Step4: Write the equation in slope - intercept form

Using the slope - intercept form $y=mx + b$, with $m=-2$ and $b = 2$, we get $y=-2x + 2$.

Step1: Recall the point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope. We are given $m=-2$ and the point $(-1,5)$, so $x_1=-1,y_1 = 5$.

Step2: Substitute into point - slope form

Substitute $m=-2,x_1=-1,y_1 = 5$ into $y - y_1=m(x - x_1)$: $y - 5=-2(x-(-1))$, which simplifies to $y - 5=-2(x + 1)$.

Step3: Convert to slope - intercept form

Expand the right - hand side: $y-5=-2x-2$. Then add 5 to both sides: $y=-2x-2 + 5$, so $y=-2x+3$.

Answer:

$y = 4x-3$

Problem 3: