Sure, let's determine the intercepts of the linear equation [tex]\(-\frac{x}{10} + \frac{y}{8} = 1\)[/tex].
Step 1: Finding the [tex]\(x\)[/tex]-intercept
The [tex]\(x\)[/tex]-intercept is found by setting [tex]\(y = 0\)[/tex] and solving for [tex]\(x\)[/tex].
1. Substitute [tex]\(y = 0\)[/tex] into the equation:
[tex]\[
-\frac{x}{10} + \frac{0}{8} = 1
\][/tex]
2. Simplify the equation:
[tex]\[
-\frac{x}{10} = 1
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = -10
\][/tex]
Therefore, the [tex]\(x\)[/tex]-intercept is [tex]\(-10\)[/tex].
Step 2: Finding the [tex]\(y\)[/tex]-intercept
The [tex]\(y\)[/tex]-intercept is found by setting [tex]\(x = 0\)[/tex] and solving for [tex]\(y\)[/tex].
1. Substitute [tex]\(x = 0\)[/tex] into the equation:
[tex]\[
-\frac{0}{10} + \frac{y}{8} = 1
\][/tex]
2. Simplify the equation:
[tex]\[
\frac{y}{8} = 1
\][/tex]
3. Solve for [tex]\(y\)[/tex]:
[tex]\[
y = 8
\][/tex]
Therefore, the [tex]\(y\)[/tex]-intercept is [tex]\(8\)[/tex].
Summary
- [tex]\(x\)[/tex]-intercept: [tex]\(-10\)[/tex]
- [tex]\(y\)[/tex]-intercept: [tex]\(8\)[/tex]
Graphing the Equation
To graph the equation, plot the intercepts on the coordinate plane and draw the line that passes through these points.
1. Plot the point [tex]\((-10, 0)\)[/tex] on the x-axis.
2. Plot the point [tex]\((0, 8)\)[/tex] on the y-axis.
3. Draw a straight line through these two points, extending in both directions.
This line is the graph of the equation [tex]\(-\frac{x}{10} + \frac{y}{8} = 1\)[/tex]. Here's a rough sketch:
[tex]\[
\begin{array}{c}
\begin{tikzpicture}
\draw[thick, ->] (-11,0) -- (1,0) node[anchor=north west] {x};
\draw[thick, ->] (0,-1) -- (0,9) node[anchor=south east] {y};
\draw[thick,] (-10.0,0) node(pin north west left: $(-10,0)$) -- (0,8.0) node(pin north east left: $(0,8)$);
\end{tikzpicture}
\end{array}
\][/tex]
With the points [tex]\((-10, 0)\)[/tex] and [tex]\((0, 8)\)[/tex], our line will intersect the axes at the correct intercepts as calculated.
