Let's find the equation of the line that passes through the points [tex]\((-1, 5)\)[/tex] and [tex]\((1, -5)\)[/tex].
1. Calculate the slope (m) of the line:
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the points [tex]\((-1, 5)\)[/tex] and [tex]\((1, -5)\)[/tex]:
[tex]\[
m = \frac{-5 - 5}{1 - (-1)} = \frac{-10}{2} = -5
\][/tex]
2. Find the y-intercept (b) of the line:
Using the point-slope form of the equation of a line:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
We can rearrange this into the slope-intercept form:
[tex]\[
y = mx + b
\][/tex]
Let's use the point [tex]\((-1, 5)\)[/tex]:
[tex]\[
5 = (-5)(-1) + b
\][/tex]
Simplify and solve for [tex]\(b\)[/tex]:
[tex]\[
5 = 5 + b \implies b = 0
\][/tex]
Therefore, the equation of the line in slope-intercept form is:
[tex]\[
y = -5x + 0 \quad \text{or simply} \quad y = -5x
\][/tex]
3. Compare with the given equations:
The given options for the equations are:
[tex]\[
\begin{array}{l}
y = -5x \\
y = 5x + 10 \\
y = -2x + 3
\end{array}
\][/tex]
The equation that matches our derived equation [tex]\(y = -5x\)[/tex] is the first one [tex]\(y = -5x\)[/tex].
Therefore, the correct linear equation that passes through the points [tex]\((-1, 5)\)[/tex] and [tex]\((1, -5)\)[/tex] is:
[tex]\[
y = -5x
\][/tex]
