Which of the following linear equations passes through points [tex]\((2,0)\)[/tex] and [tex]\((5,-3)\)[/tex]?

A. [tex]\(y = x + 2\)[/tex]
B. None of these choices are correct.
C. [tex]\(y = -x + 2\)[/tex]
D. [tex]\(y = -x - 2\)[/tex]
E. [tex]\(y = x - 2\)[/tex]



Answer :

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]