To determine the equation of the line passing through the origin [tex]\((0, 0)\)[/tex] and the point [tex]\((-5, 6)\)[/tex], we can follow these steps:
1. Identify the Points: We have two points:
- Point 1: Origin [tex]\((0, 0)\)[/tex]
- Point 2: [tex]\((-5, 6)\)[/tex]
2. Calculate the Slope:
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]
Substituting the coordinates of our points [tex]\((0, 0)\)[/tex] and [tex]\((-5, 6)\)[/tex], we get:
[tex]\[
m = \frac{6 - 0}{-5 - 0} = \frac{6}{-5} = -\frac{6}{5}
\][/tex]
3. Equation of the Line:
The general form of the equation of a line through the origin [tex]\(y = mx\)[/tex]. Using the slope calculated, the equation of our line becomes:
[tex]\[
y = -\frac{6}{5}x
\][/tex]
4. Compare with Given Options:
The equation we derived is:
[tex]\[
y = -\frac{6}{5}x
\][/tex]
Comparing this with the given options:
- A. [tex]\(y=-\frac{6}{5} x\)[/tex]
- B. [tex]\(y=-6 x\)[/tex]
- C. [tex]\(y=-\frac{5}{6} x\)[/tex]
- D. [tex]\(y=-5 x\)[/tex]
We see that option A matches our derived equation.
Therefore, the correct answer is:
A. [tex]\(y=-\frac{6}{5} x\)[/tex]
