Sure, I'd be happy to help explain the solution to this problem!
To find the equation of the line that passes through the points [tex]\((-5, -1)\)[/tex] and [tex]\( (5, 5)\)[/tex], we'll start by following a step-by-step process to determine the line's slope and y-intercept.
### Step 1: Calculate the Slope (m)
The slope [tex]\( m \)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, the points are [tex]\((-5, -1)\)[/tex] and [tex]\( (5, 5)\)[/tex]. Plugging these values into the formula, we get:
[tex]\[ x_1 = -5, \ y_1 = -1 \][/tex]
[tex]\[ x_2 = 5, \ y_2 = 5 \][/tex]
So,
[tex]\[ m = \frac{5 - (-1)}{5 - (-5)} = \frac{5 + 1}{5 + 5} = \frac{6}{10} = \frac{3}{5} \][/tex]
### Step 2: Find the Y-Intercept (b)
The y-intercept [tex]\( b \)[/tex] can be found using the slope-intercept form of the equation [tex]\( y = mx + b \)[/tex]. We can rearrange this to solve for [tex]\( b \)[/tex]:
[tex]\[ b = y - mx \][/tex]
We already have the slope [tex]\( m = \frac{3}{5} \)[/tex]. Now, we can use one of the given points to find [tex]\( b \)[/tex]. Let's use the point [tex]\((-5, -1)\)[/tex]:
[tex]\[ y = -1, \ x = -5 \][/tex]
Now substitute [tex]\( m \)[/tex], [tex]\( x \)[/tex], and [tex]\( y \)[/tex] into the equation:
[tex]\[ b = -1 - \left( \frac{3}{5} \times (-5) \right) \][/tex]
[tex]\[ b = -1 - \left( -3 \right) \][/tex]
[tex]\[ b = -1 + 3 \][/tex]
[tex]\[ b = 2 \][/tex]
### Step 3: Form the Equation of the Line
Now that we have both the slope [tex]\( m = \frac{3}{5} \)[/tex] and the y-intercept [tex]\( b = 2 \)[/tex], we can write the equation of the line in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y = \frac{3}{5}x + 2 \][/tex]
### Selecting the Correct Answer
Given the options:
- A. [tex]\( y = \frac{3}{5}x - 8 \)[/tex]
- B. [tex]\( y = \frac{2}{5}x + 3 \)[/tex]
- C. [tex]\( y = \frac{3}{5}x + 2 \)[/tex]
- D. [tex]\( y = \frac{2}{5}x - 7 \)[/tex]
The correct answer is:
[tex]\[ \boxed{C. \ y = \frac{3}{5}x + 2} \][/tex]
So, the equation of the line that passes through the points [tex]\((-5, -1)\)[/tex] and [tex]\( (5, 5)\)[/tex] is [tex]\( y = \frac{3}{5}x + 2 \)[/tex].
