To solve the given problem, we need to find the x-intercept, y-intercept, and slope of the linear equation [tex]\(10x + 12y = 240\)[/tex].
Step-by-Step Detailed Solution:
1. Finding the x-intercept:
To find the x-intercept, we set [tex]\(y = 0\)[/tex] in the equation and solve for [tex]\(x\)[/tex].
[tex]\[
10x + 12(0) = 240
\][/tex]
Simplifying this equation:
[tex]\[
10x = 240
\][/tex]
Divide both sides by 10:
[tex]\[
x = 24
\][/tex]
Therefore, the x-intercept is at [tex]\((24, 0)\)[/tex].
2. Finding the y-intercept:
To find the y-intercept, we set [tex]\(x = 0\)[/tex] in the equation and solve for [tex]\(y\)[/tex].
[tex]\[
10(0) + 12y = 240
\][/tex]
Simplifying this equation:
[tex]\[
12y = 240
\][/tex]
Divide both sides by 12:
[tex]\[
y = 20
\][/tex]
Therefore, the y-intercept is at [tex]\((0, 20)\)[/tex].
3. Finding the slope:
We rearrange the given equation into the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
The given equation is:
[tex]\[
10x + 12y = 240
\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[
12y = -10x + 240
\][/tex]
Divide every term by 12:
[tex]\[
y = -\frac{10}{12}x + \frac{240}{12}
\][/tex]
Simplify the fractions:
[tex]\[
y = -\frac{5}{6}x + 20
\][/tex]
Therefore, the slope of the line is [tex]\(-\frac{5}{6}\)[/tex].
So, the final answers are:
- The x-intercept is located at [tex]\((24, 0)\)[/tex].
- The y-intercept is located at [tex]\((0, 20)\)[/tex].
- The slope of the line represented by this equation is [tex]\(-\frac{5}{6}\)[/tex].
