To find the equation of the line that passes through the points [tex]\((5, -3)\)[/tex] and [tex]\((-10, 15)\)[/tex], we follow these steps:
### 1. Calculate the Slope (m)
The slope of a line through 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]
Using the given points [tex]\((5, -3)\)[/tex] and [tex]\((-10, 15)\)[/tex]:
[tex]\[ m = \frac{15 - (-3)}{-10 - 5} \][/tex]
[tex]\[ m = \frac{15 + 3}{-10 - 5} \][/tex]
[tex]\[ m = \frac{18}{-15} \][/tex]
[tex]\[ m = -\frac{6}{5} \][/tex]
### 2. Calculate the Y-Intercept (b)
The equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
We can use one of the points and the slope to find [tex]\(b\)[/tex]. Let's use the point [tex]\((5, -3)\)[/tex]:
[tex]\[ -3 = -\frac{6}{5}(5) + b \][/tex]
[tex]\[ -3 = -6 + b \][/tex]
[tex]\[ b = -3 + 6 \][/tex]
[tex]\[ b = 3 \][/tex]
### 3. Write the Equation of the Line
Now that we have the slope [tex]\(m = -\frac{6}{5}\)[/tex] and the y-intercept [tex]\(b = 3\)[/tex], the equation of the line is:
[tex]\[ y = -\frac{6}{5} x + 3 \][/tex]
Thus, the correct answer is:
[tex]\[ y = -\frac{6}{5} x + 3 \][/tex]
This corresponds to the given choice:
[tex]\[ y = -\frac{6}{5} x + 3 \][/tex]
So, the correct answer is:
[tex]\[ y = -\frac{6}{5} x + 3 \][/tex]
