Answer :
To find the equation of a line passing through two points, you need to follow several steps which involve calculating the slope of the line and then using the slope to find the y-intercept. We will work through these steps using the given points [tex]\((0, 4)\)[/tex] and [tex]\((-3, -11)\)[/tex].
### Step 1: Calculate the slope (m) of the line
The formula for the slope [tex]\(m\)[/tex] between 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]
Substitute the given points into the formula:
[tex]\[ (x_1, y_1) = (0, 4) \][/tex]
[tex]\[ (x_2, y_2) = (-3, -11) \][/tex]
Now, plug these values into the slope formula:
[tex]\[ m = \frac{-11 - 4}{-3 - 0} \][/tex]
[tex]\[ m = \frac{-15}{-3} \][/tex]
[tex]\[ m = 5 \][/tex]
### Step 2: Use the point-slope form to find the equation
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using point [tex]\((x_1, y_1) = (0, 4)\)[/tex] and the slope [tex]\(m = 5\)[/tex], we substitute these values into the point-slope form:
[tex]\[ y - 4 = 5(x - 0) \][/tex]
Simplifying this equation:
[tex]\[ y - 4 = 5x \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex] to get the slope-intercept form
To express the equation in the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 4 = 5x \][/tex]
Add 4 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = 5x + 4 \][/tex]
### Conclusion
Therefore, the equation of the line passing through the points [tex]\((0, 4)\)[/tex] and [tex]\((-3, -11)\)[/tex] is:
[tex]\[ y = 5x + 4 \][/tex]
### Step 1: Calculate the slope (m) of the line
The formula for the slope [tex]\(m\)[/tex] between 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]
Substitute the given points into the formula:
[tex]\[ (x_1, y_1) = (0, 4) \][/tex]
[tex]\[ (x_2, y_2) = (-3, -11) \][/tex]
Now, plug these values into the slope formula:
[tex]\[ m = \frac{-11 - 4}{-3 - 0} \][/tex]
[tex]\[ m = \frac{-15}{-3} \][/tex]
[tex]\[ m = 5 \][/tex]
### Step 2: Use the point-slope form to find the equation
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using point [tex]\((x_1, y_1) = (0, 4)\)[/tex] and the slope [tex]\(m = 5\)[/tex], we substitute these values into the point-slope form:
[tex]\[ y - 4 = 5(x - 0) \][/tex]
Simplifying this equation:
[tex]\[ y - 4 = 5x \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex] to get the slope-intercept form
To express the equation in the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 4 = 5x \][/tex]
Add 4 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = 5x + 4 \][/tex]
### Conclusion
Therefore, the equation of the line passing through the points [tex]\((0, 4)\)[/tex] and [tex]\((-3, -11)\)[/tex] is:
[tex]\[ y = 5x + 4 \][/tex]
