To find the equation of the line passing through the points [tex]\((4, -5)\)[/tex] and [tex]\((1, 4)\)[/tex], we need to determine two key components: the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex].
### Step 1: 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]
Substitute the coordinates [tex]\((x_1, y_1) = (4, -5)\)[/tex] and [tex]\((x_2, y_2) = (1, 4)\)[/tex]:
[tex]\[
m = \frac{4 - (-5)}{1 - 4} = \frac{4 + 5}{1 - 4} = \frac{9}{-3} = -3
\][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(-3\)[/tex].
### Step 2: Calculate the Y-Intercept
The equation of a line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
To find the y-intercept [tex]\(b\)[/tex], we can use one of the given points and the calculated slope. We'll use the point [tex]\((4, -5)\)[/tex].
Substitute [tex]\(m = -3\)[/tex], [tex]\(x = 4\)[/tex], and [tex]\(y = -5\)[/tex] into the equation [tex]\(y = mx + b\)[/tex]:
[tex]\[
-5 = -3(4) + b
\][/tex]
Simplify and solve for [tex]\(b\)[/tex]:
[tex]\[
-5 = -12 + b
\][/tex]
Add 12 to both sides:
[tex]\[
-5 + 12 = b
\][/tex]
[tex]\[
b = 7
\][/tex]
So, the y-intercept [tex]\(b\)[/tex] is [tex]\(7\)[/tex].
### Step 3: Write the Equation
Now that we have the slope [tex]\(m = -3\)[/tex] and the y-intercept [tex]\(b = 7\)[/tex], we can write the equation of the line:
[tex]\[
y = -3x + 7
\][/tex]
Thus, the equation of the line passing through the points [tex]\((4, -5)\)[/tex] and [tex]\((1, 4)\)[/tex] is:
[tex]\[
y = -3x + 7
\][/tex]
