Download Solving Linear Equations with Addition and Subtraction

Solving Linear Equations
Objective: I will Use properties of numbers to solve
linear equations with rational number
coefficients
Warm Up
Add, subtract, multiply, or divide.
1. 24 + 17 41
3. 12
3
–64
5.
8
2. 23 – 19
4
36
4. 6(–7)
–8
6. –250 + (–85) –335
–42
Equation
• mathematical sentence that uses an equal sign to
show that two expressions have the same value
Addition Property of Equality
•
Add the same number to each side of the equals sign
If a = b, then a + c = b + c
x–5=2
Subtraction Property of Equality
• Subtract the same number to each side of the
equals sign.
If a = b, then a – c = b – c
x+2=7
Inverse Operations
Addition and subtraction are inverse operations,
which means they “undo” each other.
* To solve an equation, use inverse operations to isolate the
variable. In other words, get the variable alone on one side of
the equal sign.
Example 1: Determining Whether a Number is a Solution of an
Equation
Determine which value of x is a solution of the equation.
x + 8 = 15; x = 5, 7, or 23
Substitute each value for x in the equation.
?
x + 8 = 15
?
5 + 8 = 15
?
13=
15 
So 5 is not solution.
Substitute 5 for x.
Example 1 Continued
Determine which value of x is a solution of the
equation.
x + 8 = 15; x = 5, 7, or 23
Substitute each value for x in the equation.
?
x + 8 = 15
?
7 + 8 = 15
?
15=
15 
Substitute 7 for x.
So 7 is a solution.
Example 1 Continued
Determine which value of x is a solution of the
equation.
x + 8 = 15; x = 5, 7, or 23
Substitute each value for x in the equation.
?
x + 8 = 15
?
23 + 8 = 15
?
31=
15 
Substitute 23 for x.
So 23 is not a solution.
Solving Equations Using Addition and Subtraction
Properties
Solve.
10 + n = 18
10 + n = 18
–10
–10
0+n= 8
n= 8
Check
Use the Subtraction Property of
Equality: Subtract 10 from both
sides.
Identity Property of Zero: 0 + n = n.
10 + n = 18
?
Substitute 8 for n.
10 + 8 = 18
?
18 = 18 
Solving Equations Using Addition and Subtraction
Properties
Solve.
p–8=9
p–8=9
+8 +8
p + 0 = 17
p = 17
Check
p–8=9
?
17 – 8 = 9
?
9 = 9
Use the Addition Property of
Equality: Add 8 to both sides.
Identity Property of Zero: p + 0 = p.
Substitute 17 for p.
Solving Equations Using Addition and Subtraction
Properties
Solve.
22 = y – 11
22 = y – 11
+ 11
+ 11
Use the Addition Property of
Equality: Add 11 to both sides.
33 = y + 0
Identity Property of Zero: y + 0 = y.
33 = y
Check
22 = y – 11
?
22 = 33 – 11 Substitute 33 for y.
?
22 = 22 
Problem Solving Application
Jan and Alex are arguing over who
gets to play a board game. If Jan, on
the right, pulls with a force of 14 N,
what force is Alex exerting on the
game if the net force is 3 N?
Helpful Hint!
Force is measured in newtons (N). The number of
newtons tells the size of the force and the sign
tells the direction. Positive is to the right, and
negative is to the left.
1
Understand the Problem
The answer is the force that Alex, on the
left, is exerting on the board game.
List the important information:
• Jan, on the right pulls with a force of 14 N.
• The net force is 3 N.
Show the relationship or the information:
Net force
=
Jan’s force
+
Alex’s force
2
Make a Plan
Write an equation and solve it. Let f represent
Alex’s force on the board game, and use the
3 = f + 14
equation model.
3
Solve
3 = f + 14
– 14
– 14
Subtract 14 from both sides.
–11 = f
Alex was exerting a force of –11 N on the board
game.
4
Look Back
Check the answer by using a number
line. Move 14 units right to show Jan's
force. Move 11 units to the left to show
Alex's force.
11
14
0 1 2
18
3 4
5
6 7
8
9 10 11 12 13 14 15 16 17
Multiplication Property of Equality
• Multiply each side by the same nonzero
number.
If a = b and c ≠ 0, then a ∙ c = b ∙ c
Division Property of Equality
• Divide each side by the same nonzero number.
If a = b, and c ≠ 0, then a ÷ c = b ÷ c
5x = 30
Solve.
Solve.
More Practice
Solve.
Solve.
Solve.
Solve.
Solve these multi-step equations.
Solve.
Example 2
During one shift, a waiter earns wages of $30 and gets an
additional 15% in tips on customers’ food bills. The waiter
earns $105. What is the total of the customers’ food bills?
Guided Practice
A real estate agent’s base salary is $22,000 per year.
The agent earns a 4% commission on total sales. How
much must the agent sell to earn $60,000 in one year?
Example 5
It takes you 8 minutes to wash a car and it takes a
friend 6 minutes to wash a car. How long does it
take the two of you to wash 7 cars if you work
together?
Homework