To determine the equation of the line that passes through the points [tex]\((7, -9)\)[/tex] and [tex]\((4, 3)\)[/tex], we need to follow these steps:
1. Calculate the Slope (m):
The formula to find the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points [tex]\((7, -9)\)[/tex] and [tex]\((4, 3)\)[/tex]:
[tex]\[
m = \frac{3 - (-9)}{4 - 7} = \frac{3 + 9}{4 - 7} = \frac{12}{-3} = -4
\][/tex]
Therefore, the slope [tex]\(m\)[/tex] is [tex]\(-4\)[/tex].
2. Find the y-intercept (b):
We use the slope-intercept form of the line equation, [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept. We already have [tex]\(m = -4\)[/tex]. Now we need to find [tex]\(b\)[/tex].
We can use one of the points to solve for [tex]\(b\)[/tex]. Let's use the point [tex]\((7, -9)\)[/tex]:
[tex]\[
y = mx + b
\][/tex]
Substituting in [tex]\(x = 7\)[/tex], [tex]\(y = -9\)[/tex], and [tex]\(m = -4\)[/tex]:
[tex]\[
-9 = -4(7) + b
\][/tex]
[tex]\[
-9 = -28 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = -9 + 28 = 19
\][/tex]
Therefore, the y-intercept [tex]\(b\)[/tex] is [tex]\(19\)[/tex].
3. Write the Equation:
Now that we have the slope [tex]\(m = -4\)[/tex] and the y-intercept [tex]\(b = 19\)[/tex], we can write the equation of the line in slope-intercept form:
[tex]\[
y = -4x + 19
\][/tex]
So, the equation of the line that contains the points [tex]\((7, -9)\)[/tex] and [tex]\((4, 3)\)[/tex] is:
[tex]\[
y = -4x + 19
\][/tex]
