To determine which of the given linear equations passes through the points [tex]\((2, 0)\)[/tex] and [tex]\((5, -3)\)[/tex], we need to find and analyze the correct equation of the line that passes through these two points. Let's follow these steps:
1. Calculate the slope of the line:
The slope [tex]\( m \)[/tex] of a line passing 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]
For the points [tex]\((2, 0)\)[/tex] and [tex]\((5, -3)\)[/tex]:
[tex]\[
m = \frac{-3 - 0}{5 - 2} = \frac{-3}{3} = -1
\][/tex]
2. Find the y-intercept ([tex]\(b\)[/tex]):
We can use the slope-intercept form of the equation of a line, [tex]\( y = mx + b \)[/tex], using one of the points and the slope we just calculated. We use the point [tex]\((2, 0)\)[/tex]:
[tex]\[
0 = -1(2) + b
\][/tex]
[tex]\[
0 = -2 + b
\][/tex]
[tex]\[
b = 2
\][/tex]
Therefore, the equation of the line is:
[tex]\[
y = -x + 2
\][/tex]
3. Verify the correctness of the calculated equation:
We will check if the calculated equation passes through the second point [tex]\((5, -3)\)[/tex]:
[tex]\[
y = -5 + 2 = -3
\][/tex]
Since both points [tex]\((2, 0)\)[/tex] and [tex]\((5, -3)\)[/tex] satisfy the equation [tex]\( y = -x + 2 \)[/tex], we confirm this is the correct equation.
The correct answer is:
[tex]\[
\boxed{y = -x + 2}
\][/tex]
