Answer :
To determine the equation of the line passing through the points [tex]\((4, 15)\)[/tex] and [tex]\((-2, -21)\)[/tex], we need to find the slope and the y-intercept of the line, putting our answer in the slope-intercept form, [tex]\(y = mx + b\)[/tex].
### Step 1: Calculate the Slope (m)
The slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the coordinates of the points [tex]\((4, 15)\)[/tex] and [tex]\((-2, -21)\)[/tex]:
[tex]\[ m = \frac{-21 - 15}{-2 - 4} \][/tex]
[tex]\[ m = \frac{-36}{-6} \][/tex]
[tex]\[ m = 6 \][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(6\)[/tex].
### Step 2: Calculate the Y-Intercept (b)
We use the slope-intercept form of the equation of a line, [tex]\(y = mx + b\)[/tex], and one of the points to solve for the y-intercept [tex]\(b\)[/tex]. Using the point [tex]\((4, 15)\)[/tex]:
[tex]\[ 15 = 6 \cdot 4 + b \][/tex]
[tex]\[ 15 = 24 + b \][/tex]
[tex]\[ b = 15 - 24 \][/tex]
[tex]\[ b = -9 \][/tex]
So, the y-intercept [tex]\(b\)[/tex] is [tex]\(-9\)[/tex].
### Step 3: Write the Equation
Now that we have the slope [tex]\(m = 6\)[/tex] and the y-intercept [tex]\(b = -9\)[/tex], we can write the equation of the line in the slope-intercept form:
[tex]\[ y = 6x - 9 \][/tex]
Therefore, the equation of the line passing through the points [tex]\((4, 15)\)[/tex] and [tex]\((-2, -21)\)[/tex] is:
[tex]\[ y = 6x - 9 \][/tex]
### Step 1: Calculate the Slope (m)
The slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the coordinates of the points [tex]\((4, 15)\)[/tex] and [tex]\((-2, -21)\)[/tex]:
[tex]\[ m = \frac{-21 - 15}{-2 - 4} \][/tex]
[tex]\[ m = \frac{-36}{-6} \][/tex]
[tex]\[ m = 6 \][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(6\)[/tex].
### Step 2: Calculate the Y-Intercept (b)
We use the slope-intercept form of the equation of a line, [tex]\(y = mx + b\)[/tex], and one of the points to solve for the y-intercept [tex]\(b\)[/tex]. Using the point [tex]\((4, 15)\)[/tex]:
[tex]\[ 15 = 6 \cdot 4 + b \][/tex]
[tex]\[ 15 = 24 + b \][/tex]
[tex]\[ b = 15 - 24 \][/tex]
[tex]\[ b = -9 \][/tex]
So, the y-intercept [tex]\(b\)[/tex] is [tex]\(-9\)[/tex].
### Step 3: Write the Equation
Now that we have the slope [tex]\(m = 6\)[/tex] and the y-intercept [tex]\(b = -9\)[/tex], we can write the equation of the line in the slope-intercept form:
[tex]\[ y = 6x - 9 \][/tex]
Therefore, the equation of the line passing through the points [tex]\((4, 15)\)[/tex] and [tex]\((-2, -21)\)[/tex] is:
[tex]\[ y = 6x - 9 \][/tex]
