To find the equation of a line passing through the points [tex]\((-1, -2)\)[/tex] and [tex]\((3, 3)\)[/tex], we need to determine the slope and the y-intercept.
### Step 1: Calculate the slope ([tex]\(m\)[/tex])
The slope [tex]\(m\)[/tex] of a line passing through points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be found using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points:
[tex]\[
m = \frac{3 - (-2)}{3 - (-1)} = \frac{5}{4} = 1.25
\][/tex]
### Step 2: Find the y-intercept ([tex]\(b\)[/tex])
The y-intercept [tex]\(b\)[/tex] can be found using the point-slope form of the equation of the line, [tex]\(y = mx + b\)[/tex]. Let's use the point [tex]\((x_1, y_1) = (-1, -2)\)[/tex]:
[tex]\[
-2 = 1.25(-1) + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
-2 = -1.25 + b \quad \implies \quad b = -2 + 1.25 = -0.75
\][/tex]
So, the equation of the line is:
[tex]\[
y = 1.25x - 0.75
\][/tex]
### Step 3: Find the x-intercept
The x-intercept occurs when [tex]\(y = 0\)[/tex]. Setting [tex]\(y = 0\)[/tex] in the equation of the line:
[tex]\[
0 = 1.25x - 0.75
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
0.75 = 1.25x \quad \implies \quad x = \frac{0.75}{1.25} = 0.6
\][/tex]
### Summary
- Slope (m): [tex]\(1.25\)[/tex]
- y-intercept (b): [tex]\(-0.75\)[/tex]
- x-intercept: [tex]\(0.6\)[/tex]
Thus, the x-intercept of the line is [tex]\(0.6\)[/tex] and the y-intercept is [tex]\(-0.75\)[/tex].
