How do you find the equation of a line in standard form (ax+by=c where a, b, and c are integers) for the line passing through (1,5) and (-3,1)?
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Two-point form of equation:
y - y1 = [ ( y2 - y1 ) * ( x - x1 ) ] / ( x2 - x1 )
In this case:
x1 = 1
x2 = -3
y1 = 5
y2 = 1
y - 5 = [( 1 - 5 ) * ( x - 1 )] / [ ( -3 ) - 1 ) ]
y - 5 = ( -4 ) * ( x - 1 )/ ( -4 )
y -5 = ( - 4 / -4 ) * ( x - 1 )
y - 5 = 1 * ( x - 1 )
y - 5 = x - 1
y - x = -1 + 5
y - x = 4
Ax + By = C
A = 1
B = -1
C = 4
To find the equation of a line in standard form using the given two points, you can follow these steps:
Step 1: Calculate the slope (m) of the line using the formula: m = (y2 - y1) / (x2 - x1).
In this case, the coordinates of the two points are: (x1, y1) = (1, 5) and (x2, y2) = (-3, 1).
So, the slope (m) can be calculated as:
m = (1 - 5) / (-3 - 1) = -4 / -4 = 1.
Step 2: With the slope (m), plug it into the point-slope form of the equation of a line:
y - y1 = m(x - x1).
Using the coordinates (x1, y1) = (1, 5):
y - 5 = 1(x - 1).
Step 3: Simplify the equation obtained in step 2.
y - 5 = x - 1.
Step 4: Rewrite the equation in standard form, ax + by = c.
Rearrange the equation to have all terms on one side:
y - x = 5 - 1.
To eliminate the negative sign before x, multiply the entire equation by -1:
-x + y = 4.
Finally, rearrange the equation in standard form, where coefficients a, b, and c are integers:
x - y = -4.
So, the equation of the line passing through the points (1,5) and (-3,1) in standard form is x - y = -4.