Question

Write in slope-intercept form an equation of the line that passes through the given points.

(6,8),(3,−9)

Answer 1

To write the equation of the line in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept, we need to determine the slope first and then use one of the points to find the y-intercept.

Step 1: Find the slope (m)
The slope of a line passing through two points, (x1, y1) and (x2, y2), can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

Given our points (6, 8) and (3, -9), we can substitute them into our formula:

x1 = 6, y1 = 8
x2 = 3, y2 = -9

m = (-9 - 8) / (3 - 6)
m = (-17) / (-3)
m = 17/3
m = 5.666666666666667 (rounded to 5.67 for simplicity)

Step 2: Find the y-intercept (b) using one of the points
Next, we can use the point-slope form of a line equation, which is:

y - y1 = m(x - x1)

We will use the first point (6, 8) and the slope m = 5.67 that we calculated.

Substitute the point and the slope into the point-slope form equation:

8 - y1 = 5.67(6 - x1)

Since we know that (x1, y1) is (6, 8), this simplifies to:

8 - 8 = 5.67(6 - 6)
b = 8 - 5.67(6)

Now, we do the multiplication and subtraction:

b = 8 - 34.02
b = -26

So now we have our y-intercept, which is b = -26.

Step 3: Write the equation in slope-intercept form
Now that we have both m and b, we can write the equation of the line:

y = mx + b

Substitute m and b into the equation:

y = 5.67x - 26

This is the equation of the line in slope-intercept form that passes through the points (6, 8) and (3, -9). For exact calculations, you may want to use the more precise value of the slope (5.666666666666667) in the equation:

y = 5.666666666666667x - 26

By doing so, we obtain a more accurate representation of the line's equation.

Answer 2

Key Concepts

• Linear equations in slope-intercept form given two points

Slope-intercept form:
`y=mx+b`

• m = slope
• b = y-intercept

Slope formula:
`m=(y_2-y_1)/(x_2-x_1)`

• 
  `(x_1,y_1)` and
  `(x_2,y_2)` are two points that fall on the line

Solving the Question

We're given:

• The line passes through (6,8), (3,-9)

1) First, find the slope using the slope formula:

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

`m=(8-(-9))/(6-3)\\\\m=(8+9)/(6-3)\\\\m=(17)/(3)`

Therefore, the slope of this line (m) is
`(17)/(3)`. Plug this into slope-intercept form:

`y=(17)/(3)x+b`

2) Now, find the y-intercept by using one of the given points:

`y=(17)/(3)x+b`

Plug in one of the given points as (x,y):

`8=(17)/(3)(6)+b\\\\8=34+b\\b=8-34\\b=-26`

Therefore, the y-intercept of the line is -26. Plug this into our original equation:

`y=(17)/(3)x+b\\\\y=(17)/(3)x-26`

Answer

`y=(17)/(3)x-26`