To determine which graph represents the line [tex]\(10x + 5y = 20\)[/tex], we need to find key characteristics of the line, including its x-intercept, y-intercept, and the slope.
1. Finding the x-intercept:
The x-intercept is the point where the line crosses the x-axis. This occurs when [tex]\(y = 0\)[/tex].
Substitute [tex]\(y = 0\)[/tex] into the equation:
[tex]\[
10x + 5(0) = 20
\][/tex]
Simplify to find [tex]\(x\)[/tex]:
[tex]\[
10x = 20
\][/tex]
[tex]\[
x = 2
\][/tex]
Thus, the x-intercept is [tex]\((2, 0)\)[/tex].
2. Finding the y-intercept:
The y-intercept is the point where the line crosses the y-axis. This occurs when [tex]\(x = 0\)[/tex].
Substitute [tex]\(x = 0\)[/tex] into the equation:
[tex]\[
10(0) + 5y = 20
\][/tex]
Simplify to find [tex]\(y\)[/tex]:
[tex]\[
5y = 20
\][/tex]
[tex]\[
y = 4
\][/tex]
Thus, the y-intercept is [tex]\((0, 4)\)[/tex].
3. Finding the slope:
The slope of the line can be derived from rewriting the equation in the slope-intercept form ([tex]\(y = mx + b\)[/tex]), where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
Rearrange the equation [tex]\(10x + 5y = 20\)[/tex] to the slope-intercept form:
[tex]\[
10x + 5y = 20
\][/tex]
Subtract [tex]\(10x\)[/tex] from both sides:
[tex]\[
5y = -10x + 20
\][/tex]
Divide each term by 5:
[tex]\[
y = -2x + 4
\][/tex]
From the equation [tex]\(y = -2x + 4\)[/tex], we identify the slope [tex]\(m\)[/tex] as [tex]\(-2\)[/tex].
4. Summary of Characteristics:
- x-intercept: [tex]\((2, 0)\)[/tex]
- y-intercept: [tex]\((0, 4)\)[/tex]
- Slope: [tex]\(-2\)[/tex]
Therefore, the graph representing the line [tex]\(10x + 5y = 20\)[/tex] will cross the x-axis at [tex]\((2, 0)\)[/tex] and the y-axis at [tex]\((0, 4)\)[/tex], and it will have a slope of [tex]\(-2\)[/tex], meaning the line will decline steeply downwards from left to right.
The line effectively captures these characteristics when graphically plotted.
