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 (x1, 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