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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=3
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
5
x

5
= -9

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:
1. 3x + 6 = 12
2. 4.5 = 3 + 2x
3. -2x + 2 - 4x = 20
4. 4(x - 2) = -10
x x 1
5.  
6
8
8
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 3x  5  8x  8x  30  8x
5  5x  30
35  5x
7  x



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
y  b  mx
y b
x
m



Solve for l
2w +2l =P
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
C(-5,0)
B