To find the equation of the line passing through the points (1, 8) and (-2, 5), we can use the slope-intercept form of a line equation, which is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope of the line and [tex]\( b \)[/tex] is the y-intercept.
Step-by-Step Solution:
1. Calculate the slope (m):
The slope of a line that passes through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the points [tex]\((1, 8)\)[/tex] and [tex]\((-2, 5)\)[/tex] into the formula:
[tex]\[
m = \frac{5 - 8}{-2 - 1} = \frac{-3}{-3} = 1
\][/tex]
2. Determine the y-intercept (b):
Once we have the slope, we can use one of the points to find the y-intercept. Using the slope-intercept form [tex]\( y = mx + b \)[/tex] and substituting [tex]\( m = 1 \)[/tex] and the point [tex]\((1, 8)\)[/tex]:
[tex]\[
8 = 1(1) + b
\][/tex]
Simplifying to solve for [tex]\( b \)[/tex]:
[tex]\[
8 = 1 + b
\][/tex]
[tex]\[
b = 8 - 1 = 7
\][/tex]
3. Write the equation of the line:
Now that we have the slope ([tex]\( m = 1 \)[/tex]) and the y-intercept ([tex]\( b = 7 \)[/tex]), we can write the equation of the line:
[tex]\[
y = 1x + 7
\][/tex]
Simplifying, we get:
[tex]\[
y = x + 7
\][/tex]
So, the equation of the line that passes through the points (1, 8) and (-2, 5) is [tex]\( y = x + 7 \)[/tex].
