solving equations

1. Important Rules:
Standard Form
Ax + By = C
Rules:
• A is positive
• A,B,C are whole #s
• No fractions/decimals

2. Example 1
Convert the equation to standard form:
y = 2x + 4
-2x -2x
-2x + y = 4
-1 -1 -1
2x – y = -4
*To make A positive, divide
(or multiply) everything by -1

3. Example 2
Write an equation in standard form:
m = 3/2 b = 12
y = 3x + 12
2
-3/2x -3/2x
-3x + y = 12
2
-3x + 2y = 24
-1 -1 -1
3x – 2y = -24
(2) (2) (2)
Plug into y = mx + b
*To get rid of the fraction,
multiply everything by the
denominator.

4. Writing Equations
of a Line

5. Various Forms of an Equation of a
Line.
Slope-Intercept Form
Standard Form
Point-Slope Form
y mx b
m
b y
 

 
slope of the line
intercept
Ax  By 
C
A B C
A A
, , and are integers
0, must be postive

 
y  y  m x 
x
m

x y
1 1
slope of the line
, is any point
 
1 1

6. EXAMPLE 1 Write an equation given the slope and y-intercept
SOLUTION
3
4
From the graph, you can see that the slope is m =
and the y-intercept is b = –2. Use slope-intercept form
to write an equation of the line.
y = mx + b Use slope-intercept form.
3
4
y = x + (–2)
3
4
Substitute for m and –2 for b.
3
4
y = x (–2)
Simplify.

7. EXAMPLE 2 Write an equation given two points
Write an equation of the line that passes through
(5, –2) and (2, 10).
SOLUTION
The line passes through (x1, y1) = (5,–2) and
(x2, y2) = (2, 10). Find its slope.
y2 – y1 m =
x2 – x1
10 – (–2)
=
2 – 5
12
–3
= = –4

8. EXAMPLE 2 Write an equation given two points
You know the slope and a point on the line, so use
point-slope form with either given point to write an
equation of the line. Choose (x1, y1) = (4, – 7).
y2 – y1 = m(x – x1) Use point-slope form.
y – 10 = – 4(x – 2) Substitute for m, x1, and y1.
y – 10 = – 4x + 8 Distributive property
y = – 4x + 8 Write in slope-intercept form.

9. Steps for Solving
Two-Step Equations
1. Solve for any Addition or Subtraction on the
variable side of equation by “undoing” the
operation from both sides of the equation.
2. Solve any Multiplication or Division from variable
side of equation by “undoing” the operation from
both sides of the equation.

10. Opposite Operations
Addition  Subtraction
Multiplication  Division

11. Helpful Hints?
Identify what operations are on the
variable side. (Add, Sub, Mult, Div)
“Undo” the operation by using
opposite operations.
Whatever you do to one side, you
must do to the other side to keep
equation balanced.

12. Ex. 1: Solve 4x – 5 = 11
4x – 5 = 15
+5 +5 (Add 5 to both sides)
4x = 20 (Simplify)
4 4 (Divide both sides by 4)
x = 5 (Simplify)