Equation of a line in general form is
Ax + By + C = 0
When you write the equation of a line in general form, you have to keep everything on the left side of the equation and nothing on the right side. That is, x-term, y-term and constant on the left side and zero on the right side.
Note :
In Ax + By + C = 0, usually we keep A as positive. If A is negative, multiply both sides of the equation by -1 and make A as positive.
Examples 1-4 : Write the given equations of lines in general form :
Example 1 :
y = -2x + 3
Solution :
y = -2x + 3
Add 2x to both sides.
2x + y = 3
Subtract 3 from both sides.
2x + y - 3 = 0
Example 2 :
y = 3x - 1
Solution :
y = 3x - 1
Subtract 3x from both sides.
-3x + y = -1
Add 1 to both sides.
-3x + y + 1 = 0
Multiply both sides of the equation by -1.
3x - y - 1 = 0
Example 3 :
y = 2x/3 - 1
Solution :
y = 2x/3 - 1
Multiply both sides of the equation 3.
3(y) = 3(2x/3 - 1)
3y = 3(2x/3) + 3(-1)
3y = 2x - 3
Subtract 2x from both sides.
-2x + 3y = -3
Add 3 to both sides.
-2x + 3y + 3 = 0
Multiply both sides of the equation by -1.
2x - 3y - 3 = 0
Example 4 :
y = 5x/4 - 7/6
Solution :
y = 5x/4 - 7/6
In the equation above, we find two different denominators 4 and 6.
The least common multiple of (4, 6) = 12.
Multiply both sides of the equation by 12 to get rid of the denominators 4 and 6.
12(y) = 12(5x/4 - 7/6)
12y = 12(5x/4) + 12(-7/6)
12y = 3(5x) + 2(-7)
12y = 15x - 14
Subtract 15x from both sides.
-15x + 12y = -14
Add 14 to both sides.
-15x + 12y + 14 = 0
Multiply both sides of the equation by -1.
15x - 12y - 14 = 0
Examples 5-6 : Write the equations of the given lines in general form :
Example 5 :
Solution :
Formula to find the slope of a line when two points are given :
Substitute (x1, y1) = (-4, -4) and (x2, y2) = (5, 2).
Equation of the line in slope-intercept form :
y = mx + b
Substitute m = 2/3.
y = 2x/3 + b ----(1)
Substitute one of the two points on the line into the above equation to solve for b.
Substitute (5, 2) into the above equation.
2 = 2(5)/3 + b
2 = 10/3 + b
Subtract 10/3 from both sides.
2 - 10/3 = b
-4/3 = b
Substitute b = -4/3 in (1).
y = 2x/3 - 4/3
Multiply both sides by 3 to get rid of the denominators.
3(y) = 3(2x/3 - 4/3)
3y = 3(2x/3) + 3(-4/3)
3y = 2x - 4
Subtract 2x from both sides.
-2x + 3y = -4
Add 4 to both sides.
-2x + 3y + 4 = 0
Multiply both sides of the equation by -1.
2x - 3y - 4 = 0
Example 6 :
Solution :
Formula to find the slope of a line when two points are given :
Substitute (x1, y1) = (-4, 2) and (x2, y2) = (4, -4).
Equation of the line in slope-intercept form :
y = mx + b
Substitute m = -3/4.
y = -3x/4 + b ----(1)
Substitute one of the two points on the line into the above equation to solve for b.
Substitute (-4, 2) into the above equation.
2 = -3(-4)/4 + b
2 = -3(-1) + b
2 = 3 + b
Subtract 3 from both sides.
-1 = b
Substitute b = -1 in (1).
y = -3x/4 - 1
Multiply both sides by 4.
y = -3x/4 - 1
4(y) = 4(-3x/4 - 1)
4(y) = 4(-3x/4) + 4(-1)
4y = -3x - 4
Add 3x to both sides.
3x + 4y = -4
Add 4 to both sides.
3x + 4y + 4 = 0
Example 7 :
Write the equation of a line in general form that passes through the point (4, -1) and has slope -2.
Solution :
Equation of the line in slope-intercept form :
y = mx + b
Given : Slope is -2. So, substitute m = -2.
y = -2x + b ----(1)
Since the line passes through the point (4, -1) substitute the point (4, -1) into the above equation.
-1 = -2(4) + b
-1 = -8 + b
Add 8 to both sides.
7 = b
Substitute b = 7 in (1).
y = -2x + 7
Add 2x from both sides.
2x + y = 7
Subtract 7 from both sides.
2x + y - 7 = 0
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
Apr 28, 24 10:10 AM
Apr 28, 24 05:42 AM
Apr 27, 24 11:06 AM