Download 1.1 Tips for Success

3.1 Linear Equations: One
Transformation
Linear Equations in One Variable
A linear equation in one variable is an equation
that can be written in the form
ax + by = c, where a, b, and c are real numbers
and a ≠ 0.
An equation is a statement with an equals sign.
3.1 Linear Equations:One
Transformation
Original
Equation
Equivalent
Equation
1. Add the same number to both sides.
x-4=7
x = 11
2. Subtract the same number to both sides.
x+2=9
x=7
3. Multiply both sides by the same number.
x/2 = 4
x=8
4. Divide both sides by the same number.
5x = 20
x=4
x=7
7=x
5. Interchange the two sides.
3.1 Linear Equations:One
Transformation


Two equations are equivalent if they have the
same solution set.
In solving an equation, the goal is to isolate the
variable by using inverse operations.
3.1 Linear Equations:One
Transformation
Solve
x+3=7
 x + 3 - 3 = 7 - 3 Add -3 to both sides
 x = 4
Simplify

3.1 Linear Equations:One
Transformation
Solve
3x = -12
3x 12

Divide both sides by 3
3
3
x  4

Simplify

3.1 Linear Equations:One
Transformation
Solve
-4 = x - 7
4 + 7 = x - 7 + 7 Add 7 to both sides.
3 x
Simplify
x3
Transpose
3.1 Linear Equations:One
Transformation
Solve
x
= -9
5
x

5


5 = -9(5) Multiply both sides by 5.

x  45

Simplify
3.1 Linear Equations:One
Transformation
Properties of Equalities
1. If a = b, then a + c, = b + c Addition Property
2. If a = b, then a - c, = b - c Subtraction Property
3. If a = b, then ac = bc
Multiplication Property
a b
4. If a  b and c  0, then 
Division Property
c c
3.2 Linear Equations: Two or More
Transformations

Using Two or More Transformations
1. Simplify both sides of the equation.
2. Use inverse operations to isolate the variable.
3. Check the solution.
3.2 Linear Equations:Two or
More Transformations
Solve
4 x + 7 - 2x = 13
2x  7 13
Combine like terms.
2x  6
Subtract 7 from both sides.
x3
Divide both sides by 2.
3.2 Linear Equations:Two or
More Transformations
Solve
5x +2(3 - x) =15
5x  6  2x 15
Remove parentheses
3x  6 15
Combine like terms.
3x  9
x3
Subtract 6 from both sides.
Divide both sides by 3.
3.2 Linear Equations: Two or
More Transformations
Solve:
3x + 6 = 12
3x + 6 - 6 = 12 – 6 Add -6 to both sides
3x = 6
Combine like terms
x=2
Divide both sides by 3
3.2 Linear Equations: Two or
More Transformations
Solve:
4.5 = 3 + 2x
4.5 - 3 = 3 - 3 + 2x
1.5 = 2x
.75 = x
Add -3 to both sides
Divide both sides by 2
3.2 Linear Equations: Two or
More Transformations
Solve:
-2x + 2 - 4x = 20
-6x + 2 = 20
-6x + 2 – 2 = 20 – 2
-6x = 18
x = -3
Combine like terms
Add -2 to both sides
Divide by -6
3.2 Linear Equations: Two or
More Transformations
Solve:
4(x - 2) = -10
4x - 8 = -10
4x – 8 + 8 = -10 + 8
4x = -2
x = -2/4 = -1/2
Remove parenthesis
Add 8 to both sides
Combine like terms
Divide both sides by -4
3.2 Linear Equations: Two or
More Transformations
Solve:
-3(x - 2) = 21
-3x + 6 = 21
Remove parenthesis
-3x + 6 - 6 = -21 - 6 Add -6 to both sides
-3x = -27
Combine like terms
x =9
Divide both sides by -3
3.3 Solving Equations with
Variables on Both Sides
Collect variables on the side with the greatest variable
coefficient
3x  5  8x  30
3x  5  8x  30
3x  5  3x  8x  30  3x
5  5x  30
35  5x
7  x



3x  5  8x  8x  30  8x
5x  5  30
5x  35
x  7
3.3 Solving Equations with
Variables on Both Sides
Solve:
4( y  2)  6y  2  8y
4 y  8  2y  2
4 y  8  2y  2y  2  2y

6y  8  2

6y  8  8  2  8
6y  6
y 1
3.3 Solving Equations with
Variables on Both Sides
Solve:
3(x  5)  2x 10  4x
3x 15  2x 10
5x 15  10
5x  5
x 1
3.4 Problem Solving
General Strategies for Problem Solving
1. Read and understand the problem.
Choose a variable
Construct a diagram
2. Translate the problem into an equation.
3. Solve the equation.
4. Interpret the results.(Check your answer)
3.5 Solving Equations That Involve
Decimals
Solve:
12.3x - 5.1 = 17
12.3x - 5.1+5.1 = 17 + 5.1
12.3x = 22.1
22.1
x
12.3
x  1.796747
3.6 Literal Equations


Solve for v
d =vt
d
v
t


Solve for x
y = mx + b


Solve for l
2w +2l =P
y  b  mx


y b
x
m
2l =P - 2w
P  2w
l
2
3.7 Scatter Plots
Quadrant II
Quadrant I
Quadrant III
Quadrant IV
3.7 Scatter Plots
A(3,2)
A
B(-1,-3)
B
B
C(-5,0)