Forms of Equations
Objectives:
…to write an equation in standard form
...to write an equation in slope-intercept form
...to write an equation using function notation
Assessment Anchor:
8.C.3.1 – Plot and/or identify ordered pairs on a coordinate plane.
8.D.4.1 – Represent relationships with tables or graphs on the coordinate
plane.
NOTES
Standard Form of a linear equation:
Ax + By = C
Where A, B, and C are INTEGERS!! (no fractions or decimals allowed!)
To write a linear equation in standard form:
1. Look at the coefficients and constants
a. IF they are all integers, proceed to step 2
b. IF there are fractions, multiply every term by the LCD
c. IF there are decimals, multiply every term by a multiple of 10
2. Rearrange the equation so that:
a. The “x” term is first
b. The “y” term is second
c. The constant is on the other side of the equals sign
**** There are TWO correct (simplified) versions of the standard form of an
equation…differing by a factor of -1.
Forms of Equations
EXAMPLES
1)
y = 4x + 8
Look at the coefficients and constants ----->
Subtract 4x on both sides --------------------->
y = 4x + 8
– 4x – 4x
y – 4x = 8
Switch the “x” and “y” terms --------------->
Alternate form:
2)
-4x + y = 8
4x – y = -8
½y – 7 = 4x
Look at the coefficients and constants ----->
½y – 7 = 4x
Multiply every term by the LCD of 2 ------> (2)( ½y) – (2)(7) = (2)(4x)
1y – 14 = 8x
Subtract 8x on both sides -------------------->
– 8x – 8x
1y – 14 – 8x = 0
Add 14 on both sides ------------------------->
+ 14
+ 14
1y – 8x = 14
Switch the “x” and “y” terms --------------->
Alternate form:
3)
-3 + 3x = 7y
-8x + 1y = 14
8x – 1y = -14
Forms of Equations
– 10 = 1 y
4)
2
x
3
5)
0.3x – 4 = 0.2y
2
Look at the coefficients and constants ----->
0.3x – 4 = 0.2y
Multiply every term by 10 ------------> (10)(0.3x) – (10)(4) = (10)(0.2y)
3x – 40 = 2y
Subtract 2y on both sides -------------------->
– 2y – 2y
3x – 40 – 2y = 0
Add 40 on both sides ------------------------->
+ 40
+ 40
3x – 2y = 40
3x – 2y = 40
Alternate form:
6)
2.3y + 1.9x = 0.12
-3x + 2y = -40
Forms of Equations
MORE NOTES
Slope-Intercept Form of a linear equation:
y = mx + b
Where “m” is the slope, and “b” is the y-coordinate of the y-intercept.
To write a linear equation in slope-intercept form:
1. Solve the equation for “y”
2. Simplify the right side as far as you can
Characteristics of an equation written in slope-intercept form:
1. “y” must have a coefficient of “1”…(“y” must be by itself…)
2. “m” represents the slope
3. “b” represents the y-coordinate of the y-intercept
a. the y-intercept is the point where a line crosses the y-axis
b. its coordinates are (0,“b”)
EXAMPLES
1a)
-2y = -10 + 6x
-2y = -10 + 6x
-2
-2
***not yet correct!! ------> y = 5 – 3x
y = -3x + 5
1b) 4y = -8 + 2x
Forms of Equations
2a)
2b) 4x – 3y = 15
-2x + 4y = -20
-2x + 4y = -20
+2x
+2x
4y = -20 + 2x
4
4
***not yet correct!! ------> y = -5 + ½x
y = ½x – 5
3a)
-8y + 3x = 16
3b) 6y + 4x = 6
-8y + 3x = 16
– 3x – 3x
-8y = 16 – 3x
-8
-8
***not yet correct!! ------> y = -2 + ⅜x
y = ⅜x – 2
4a)
-2y – 8 = 3x
4b)
3x + 1 y = -1
4
Forms of Equations
MORE NOTES
Function Notation is a different way of writing equations. Basically
the “y” is now replaced with “f(x)”.
f(x) can be read: “f of x”
You might see T-tables written using function notation. They might
look like this:
1)
x
2
1
0
f(x)
3
6
9
2)
x
-1
0
1
f(x)
-3
-9
-15
You might see equations written using function notation. They might
look like this:
1)
f(x) = -3x + 9
2)
f(x) = -6x – 9
Take a moment to take this equation (function) and create a T-table
using function notation.
1)
f(x) = 5x – 7