Write the slope-intercept form of the equation of the line satisfying the given conditions.
Through (3,8); slope −4
The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope and b is the y-intercept.
Since the slope of the line is given as -4, we have m = -4.
To find b, we can substitute the coordinates (3,8) into the equation:
8 = -4(3) + b
Simplifying, we have:
8 = -12 + b
Adding 12 to both sides:
20 = b
Therefore, b = 20.
So, the slope-intercept form of the equation of the line is:
y = -4x + 20
To write the equation of a line in slope-intercept form, which is y = mx + b, we need to find the values of m and b.
We are given the slope, which is -4. The slope, m, represents the change in y divided by the change in x. So, we can say:
m = (change in y) / (change in x)
Since we are given one point on the line (3,8), and we know that (x₁,y₁) is (3,8), we can substitute the values into the equation:
-4 = (8 - y₁) / (3 - x₁)
We can simplify this expression:
-4 = (8 - y₁) / (3 - 3)
-4 = (8 - y₁) / 0
Since we have an undefined result, this means that the line is vertical and the equation is in the form x = a, where a is the x-coordinate of the point (3,8). So, the equation of the line is x = 3.
To find the slope-intercept form of the equation of a line, we need to use the given slope and a point on the line.
The slope-intercept form of a linear equation is given by:
y = mx + b
Where:
- m is the slope of the line, and
- b is the y-intercept (the point where the line crosses the y-axis).
In this case, we are given the point (3, 8) and the slope -4.
We can substitute the values into the slope-intercept form:
y = mx + b
y = -4x + b
Now, we need to find the value of b, which represents the y-intercept.
To do this, we substitute the coordinates of the given point (3, 8) into the equation:
8 = -4(3) + b
Simplifying the equation:
8 = -12 + b
To isolate b, we add 12 to both sides of the equation:
8 + 12 = b
b = 20
With the value of b, we can now write the equation of the line in slope-intercept form:
y = -4x + 20
Therefore, the slope-intercept form of the equation of the line satisfying the given conditions is y = -4x + 20.