(3,0),0,-4).solve for length, standard form.
To solve for the length and standard form of the line passing through the points (3,0) and (0,-4), we can follow these steps:
Step 1: Find the slope of the line.
The slope (m) of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
In our case, (x1, y1) = (3,0) and (x2, y2) = (0,-4).
Applying the formula, we have:
m = (-4 - 0) / (0 - 3)
m = -4 / -3
m = 4/3
Step 2: Find the length of the line.
The length of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the distance formula:
length = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In our case, (x1, y1) = (3,0) and (x2, y2) = (0,-4).
Applying the formula, we have:
length = sqrt((0 - 3)^2 + (-4 - 0)^2)
length = sqrt(9 + 16)
length = sqrt(25)
length = 5
Step 3: Write the equation in standard form.
The standard form of a line is given by the equation: Ax + By = C, where A, B, and C are integers, and A is non-negative.
To convert the equation to standard form, we start by rearranging the slope-intercept form equation y = mx + b:
y = (4/3)x - 4
Next, we multiply all the terms by the common denominator (3) to eliminate the fraction:
3y = 4x - 12
Finally, we rearrange the equation to match the standard form:
4x - 3y = 12
Therefore, the line passing through the points (3,0) and (0,-4) has a length of 5 units and can be expressed in standard form as 4x - 3y = 12.