Answer :
To find the equation of the line that passes through the points [tex]\((-3, 8)\)[/tex] and [tex]\((-2, 3)\)[/tex], we need to determine two main components: the slope and the y-intercept.
### Step 1: Calculate the Slope (m)
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 the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, [tex]\((x_1, y_1) = (-3, 8)\)[/tex] and [tex]\((x_2, y_2) = (-2, 3)\)[/tex]. Plugging in these coordinates:
[tex]\[ m = \frac{3 - 8}{-2 - (-3)} = \frac{3 - 8}{-2 + 3} = \frac{-5}{1} = -5 \][/tex]
### Step 2: Determine the Y-Intercept (b)
Once we have the slope, we can find the y-intercept by using the slope-intercept form of the equation of a line, which is [tex]\(y = mx + b\)[/tex]. We can rearrange this to solve for [tex]\(b\)[/tex]:
[tex]\[ b = y - mx \][/tex]
We can use either point to find the y-intercept. Let’s use the point [tex]\((-3, 8)\)[/tex]:
[tex]\[ b = 8 - (-5 \times -3) = 8 - 15 = -7 \][/tex]
### Step 3: Write the Equation in Slope-Intercept Form
Now that we have both the slope ([tex]\(m = -5\)[/tex]) and the y-intercept ([tex]\(b = -7\)[/tex]), we can write the equation of the line in slope-intercept form:
[tex]\[ y = -5x + (-7) \][/tex]
Simplifying the equation:
[tex]\[ y = -5x - 7 \][/tex]
Thus, the equation of the line through the points [tex]\((-3, 8)\)[/tex] and [tex]\((-2, 3)\)[/tex] is:
[tex]\[ \boxed{y = -5x - 7} \][/tex]
### Step 1: Calculate the Slope (m)
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 the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, [tex]\((x_1, y_1) = (-3, 8)\)[/tex] and [tex]\((x_2, y_2) = (-2, 3)\)[/tex]. Plugging in these coordinates:
[tex]\[ m = \frac{3 - 8}{-2 - (-3)} = \frac{3 - 8}{-2 + 3} = \frac{-5}{1} = -5 \][/tex]
### Step 2: Determine the Y-Intercept (b)
Once we have the slope, we can find the y-intercept by using the slope-intercept form of the equation of a line, which is [tex]\(y = mx + b\)[/tex]. We can rearrange this to solve for [tex]\(b\)[/tex]:
[tex]\[ b = y - mx \][/tex]
We can use either point to find the y-intercept. Let’s use the point [tex]\((-3, 8)\)[/tex]:
[tex]\[ b = 8 - (-5 \times -3) = 8 - 15 = -7 \][/tex]
### Step 3: Write the Equation in Slope-Intercept Form
Now that we have both the slope ([tex]\(m = -5\)[/tex]) and the y-intercept ([tex]\(b = -7\)[/tex]), we can write the equation of the line in slope-intercept form:
[tex]\[ y = -5x + (-7) \][/tex]
Simplifying the equation:
[tex]\[ y = -5x - 7 \][/tex]
Thus, the equation of the line through the points [tex]\((-3, 8)\)[/tex] and [tex]\((-2, 3)\)[/tex] is:
[tex]\[ \boxed{y = -5x - 7} \][/tex]
