To find the equation of the line passing through the points [tex]\((-3/4, -2)\)[/tex] and [tex]\((5/4, -3)\)[/tex], we need to determine two key components: the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex].
### Step-by-Step Solution:
1. Identify the coordinates of the given points:
[tex]\[
(x_1, y_1) = \left(-\frac{3}{4}, -2\right)
\][/tex]
[tex]\[
(x_2, y_2) = \left(\frac{5}{4}, -3\right)
\][/tex]
2. Calculate the slope [tex]\(m\)[/tex]:
The formula for the slope of a line passing 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]
Substituting the given values:
[tex]\[
m = \frac{-3 - (-2)}{\frac{5}{4} - \left(-\frac{3}{4}\right)}
\][/tex]
Simplify the numerator:
[tex]\[
m = \frac{-3 + 2}{\frac{5}{4} + \frac{3}{4}}
\][/tex]
Simplify the denominator:
[tex]\[
\frac{5}{4} + \frac{3}{4} = \frac{8}{4} = 2
\][/tex]
Now, plug in these values:
[tex]\[
m = \frac{-1}{2} = -0.5
\][/tex]
3. Calculate the y-intercept [tex]\(b\)[/tex]:
Using the slope [tex]\(m\)[/tex] and one of the given points (we can use [tex]\((x_1, y_1)\)[/tex]), we can find the y-intercept. The equation of the line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
Plugging in the values:
For the point [tex]\(\left(-\frac{3}{4}, -2\right)\)[/tex]:
[tex]\[
-2 = -0.5 \left(-\frac{3}{4}\right) + b
\][/tex]
Simplify:
[tex]\[
-2 = 0.375 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = -2.375
\][/tex]
4. Write the final equation of the line:
Now that we have both the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex], we can write the equation of the line in the form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y = -0.5x - 2.375
\][/tex]
Thus, the equation of the line passing through the points [tex]\(\left(-\frac{3}{4}, -2\right)\)[/tex] and [tex]\(\left(\frac{5}{4}, -3\right)\)[/tex] is:
[tex]\[
y = -0.5x - 2.375
\][/tex]
