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To convert from standard form to slope intercept form you just need to solve for y using the standard form of the equation.
Example #1:
Convert the equation 4x + 2y = 12 to slope-intercept form
4x + 2y = 12
Subtract 4x from each side of the equation.
4x - 4x + 2y = 12 - 4x
0 + 2y = 12 - 4x
2y = 12 - 4x
2y = 12 + -4x
2y = -4x + 12
Divide each side of the equation by 2 so the equation can be written in slope-intercept form.
2y / 2 = (-4x + 12) / 2
y = -4x / 2 + 12 / 2
y = -2x + 6
The slope-intercept form is y = mx + b and m is the slope.
Comparing y = mx + b with y = -2x + 6, we can see that the slope is m = -2
The slope of the line is -2.
Example #2:
Convert the equation of -5x + 3y = 8 to slope intercept form
-5x + 3y = 8
Add 5x to each side of the equation.
-5x + 5x + 3y = 8 + 5x
0 + 3y = 8 + 5x
3y = 5x + 8
Divide each side of the equation by 3 so the equation can be written in slope-intercept form.
3y / 3 = (5x + 8) / 3
y = 5x / 3 + 8 / 3
y = (5 / 3)x + 8/3
The slope of the line is 5 / 3.
Example #3:
Find the slope of Ax + By = C
Ax + By = C
Subtract Ax from each side of the equation.
Ax - Ax + By = C - Ax
0 + By = C - Ax
By = C - Ax
By = C + -Ax
By = -Ax + C
Divide each side of the equation by B so the equation can be written in slope-intercept form.
By / B = (-Ax + C) / B
y = -Ax / B + C / B
y = (-A / B)x + C / B
The slope of the line is -A / B.