Writing different forms of Linear equations

Linear equations are one degree equations. We can write linear equations in three ways.

Writing Linear equations in Standard form

The Standard form is ax + by = c; Example 3x + 2y = 4
We can convert linear equations in standard form to slope-intercept form y= mx+b
3x+2y=4
2y = -3x+4

y= -\frac{3}2x+2

y = mx + b, slope, m= -3/2 and y intercept, b = 2

Point-slope form

The Point-slope form of the equation passing through the point (x_1, y₁) and slope m is y – y₁= m(x – x₁)

Slope-Intercept form

The slope-intercept form the equation is y = mx + b

Standard form of equation

Standard form of equation is ax + by = c

We can easily convert point-slope form of equation to standard form and slope-intercept form.

Example: Write equation of the line passing through the point (-1, 3) and slope -2 in following three forms

a) Point slope form b) Slope-intercept form c) Standard form

a) Point slope form

y – y₁= m(x – x₁)

               y-3 = -2(x-(-1))
               y-3 = -2(x+1)

b) Slope-intercept form

Method 1

y= mx+ b

(x, y) = (-1, 3), m= -2
Substitute in y=mx+b to find b
              3 = -2(-1)+b
              3 = 2 + b
              3-2 = b
                b= 1
Equation of the line is y = -2x +1
Method 2
Use point-slope form y-3=-2(x+1)
                     y-3=-2x-2
                      +3    +3
                     y  =-2x+1

c) Standard form ax + by = c

Use slope-intercept form or point-slope form then convert to standard form
Slope-intercept form to standard form
Equation of the line passing through the point (-1,3) and slope m= -2 is y = -2x+1
y  = -2x + 1
+2x  +2x
2x+y = 1
Point slope to standard form
y - 3 = -2(x + 1)
y - 3 = -2x - 2
   +3       + 3
y     = -2x + 1
+2x     +2x
2x + y = 1

Write the following equations to Standard form ax + by = c

1.

y=\:\frac{3}{4}x+2

multiply all the terms by 4
       4y = 3x + 8
       -3x  = -3x
   -3x + 4y =  8

2. Write the following equation to standard form

y-3=\:\frac{3}{5}\left(x-1\right)

Multiply both sides of the equation by 5
5(y-3) = 3(x-1)
5y-15  = 3x - 3
  +15       + 15
5y     = 3x + 12
-3x     -3x
-3x + 5y = 12

Write the following equations to slope-intercept form

3x + 4y = 8

 3x + 4y = 8
-3x       -3x    Subtract 3x both sides of the equation.
      4y = -3x + 8
Divide all the terms by 4
       y = -3x/4 +8

2. 6x-2y = 5

 6x - 2y = 5
 -6x       -6x
     - 2y= -6x + 5
Divide all the terms by -2
        y= 3x - 5/2
        y = 3x - 2.5
3. y-3 = 2(x+2)
   y-3 = 2x + 4
    +3       +3
   y   = 2x + 7
4. x - 4y = 12
  -x        -x
     - 4y = -x + 12  divide all the terms by -4
       -4   -4   -4
   y = 1/4 (x) -3

y=\frac{1}{4}x-3