1. Solving Linear Equations
Overview

2. Solving Linear Equations
The topic of this lesson is, Solving Linear Equations
This may lead you to ask a few questions
What is an equation?
What is a linear equation?
What does it mean to solve a linear equation?
You have solved linear equations in previous math classes
But now you will see how the number properties, and a new set of properties of equality, shed light on why you solve equations they way you do

3. What is an Equation?
*If a and b are real numbers, then 𝑎=𝑏 is an equation
*This means that a and b represent the same real number
In fact, when you see “𝑎=𝑏”, it is perfectly acceptable to say either “a equals b” or “a is the same as b”
Occasionally, we will use ≠ to mean the opposite of =
That is, 𝑎≠𝑏 means that a and b are different real numbers

4. What is an Equation?
Like the Real Numbers, there are Properties of Equality that we can use to solve equations
*If 𝑎, 𝑏, and 𝑐 are any real numbers, then
𝑎=𝑎 (reflexive property)
If 𝑎=𝑏, then 𝑏=𝑎 (symmetry property)
If 𝑎=𝑏 and 𝑏=𝑐, then 𝑎=𝑐 (transitive property)
If 𝑎=𝑏, then 𝑎+𝑐=𝑏+𝑐 and 𝑎−𝑐=𝑏−𝑐 (addition property)
If 𝑎=𝑏, then 𝑎𝑐=𝑏𝑐 and 𝑎 𝑐 = 𝑏 𝑐 (as long as 𝑐≠0) (multiplication property)

5. What is an Equation?
*For our purposes, the most important properties of equality are the addition and multiplication properties
*Note that these say that we may
add (subtract) the same number from both sides of an equal sign without changing the truth of the equation
multiply (divide, except by zero) both sides of an equal sign by the same number without changing the truth of the equation

6. What is a Linear Equation?
*A linear equation is one that may be written as 𝑎𝑥+𝑏=0, where 𝑎 and 𝑏 are given numbers and 𝑥 is the unknown
If you recall from earlier math courses, an equation of a line may be written in the form 𝑦=𝑚𝑥+𝑏
A linear equation as written above has the same pattern, but with zero in place of 𝑦 and with 𝑚=𝑎
You will see later that what makes it a linear equation is the fact that the exponent on 𝑥 is 1

7. What Does it Mean to Solve a Linear Equation?
You have probably had a teacher tell you in the past that to solve an equation is to “get 𝑥 by itself”
Of maybe you were told that to solve is to “isolate 𝑥”
These two phrases get at the heart of the matter of solving a linear equation, but we would like to be more precise
Imagine that you have never solved a linear equation; that you didn’t even know what it means to solve an equation
Let’s start, then, with a definition

8. What Does it Mean to Solve a Linear Equation?
*To solve a linear equation in 𝑥 means to transform 𝑎⋅𝑥+𝑏 to 1⋅ 𝑥+0 on one side of the equal sign
Since, by the Identity Properties, 1⋅𝑥=𝑥 and 𝑥+0=𝑥, then transforming 𝑎⋅𝑥+𝑏 to 1⋅𝑥+0=𝑥 is exactly the same as “getting the 𝑥 by itself”
*The advantage of the definition is that we know how to produce 1 and how to produce 0 from our properties:
The Inverse Property for Addition: 𝑎+ −𝑎 =0
The Inverse Property for Multiplication: 𝑎⋅ 1 𝑎 =1

9. What Does it Mean to Solve a Linear Equation?
Since you are supposed to imagine that you have never solved an equation before, then this would, at first, seem mysterious:
Solve for 𝑥: 𝑥+1=10
But now you know that solving for 𝑥 means transforming the left side of the equal sign to 1⋅𝑥+0
Since we already have 𝑥=1⋅𝑥, then we need only change 1⋅𝑥+1 to 1⋅𝑥+0
You know that we get zero by adding a number to its opposite, and also that you can add to an equation as long as you add to both sides of the equal sign

10. What Does it Mean to Solve a Linear Equation?
*Then we are able to do this:
1⋅𝑥+1 =10 Given
1⋅𝑥+1 + −1 =10+ −1 Addition Property of Equations
1⋅𝑥+ 1+ −1 =10−1=9 Associative Property of Addition
1⋅𝑥+0=9 Inverse Property of Addition
𝑥=9 Definition of Solving

11. What Does it Mean to Solve a Linear Equation?
Now let’s consider this equation
Solve 1 2 ⋅𝑥=7
We need to transform the left side to 1⋅𝑥+0, but we already have 1 2 ⋅ 𝑥+0
So we need a way to produce 1⋅𝑥 where we now have 1 2 ⋅𝑥
You know we get 1 by multiplying a number by its reciprocal
You know that we get one by multiplying a number and its reciprocal, and also that you can multiply an equation as long as you multiply on both sides of the equal sign

12. What Does it Mean to Solve a Linear Equation?
*Then we are able to do this:
1 2 ⋅𝑥+0 =7 Given
2⋅ 1 2 ⋅𝑥+0 =2⋅7 Multiplication Property of Equations
2⋅ 1 2 ⋅𝑥+2⋅0=14 Distributive Property
1⋅𝑥+0=14 Inverse Property of Multiplication
𝑥= Definition of Solving

13. What Does it Mean to Solve a Linear Equation?
Now let’s use this method to solve a “2-step” equation
Solve 1 3 ⋅𝑥−7=1
We must transform the left side from 1 3 ⋅𝑥−7 to 1⋅𝑥+0
Also, it will be helpful to use our definition of subtraction and write
1 3 ⋅𝑥+ −7 =1
Also, does it matter whether we change +(−7) to +0 first, or change ⋅ to 1⋅ first?
It turns out that we can do it either way, but producing zero first is easier

14. What Does it Mean to Solve a Linear Equation?
*Then we are able to do this:
1 3 ⋅𝑥+(−7) =1 Given
1 3 ⋅𝑥+(−7) +7=1+7 Addition Property of Equations
1 3 ⋅𝑥+(−7+7)=8 Associative Property of Addition
1 3 ⋅𝑥+0=8 Inverse Property of Addition
1 3 ⋅𝑥 =8 Identity Property of Addition
3⋅ 1 3 ⋅𝑥 =3⋅8 Multiplication Property of Equations
3⋅ 1 3 ⋅𝑥=24 Associative Property of Multiplication

15. What Does it Mean to Solve a Linear Equation?
*Then we are able to do this:
3⋅ 1 3 ⋅𝑥=24 Associative Property of Multiplication
1⋅𝑥=24 Inverse Property of Multiplication
𝑥= Identity Property of Multiplication
Can we solve this same equation by first eliminating 1/3?

16. What Does it Mean to Solve a Linear Equation?
Solve for 𝑥: ⋅𝑥+ −7 =1
1 3 ⋅𝑥+ −7 =1 Given
3⋅ 1 3 ⋅𝑥+ −7 =3⋅1 Multiplication Property of Equality
3⋅ 1 3 ⋅𝑥+3⋅ −7 =3 Distributive Property
𝑥+ −21 =3 Inverse Property of Multiplication
𝑥+ − =3+21 Addition Property of Equality
𝑥+ −21+21 =24 Associative Property of Addition
𝑥= Inverse Property of Addition

17. What Does it Mean to Solve a Linear Equation?
*The solution to any equation can be represented as a solution set
For example, the solution set for the previous equation is the set with one element: 24
You will later learn to solve equations that have more than one solution
In some cases, an equation has no solution; there is no real number 𝑥 that will make the equation true
*In that case, the solution set is the empty set, represented by or by ∅

18. Terms, Like Terms, and the Distributive Property
In any expression
numbers that are added are called the terms of the expression
known values in a term are called the coefficients of the term
numbers not multiplying a variable are called constants
For example, the expression 5𝑥−3𝑦+6𝑧−1 has four terms (since subtraction is really addition)
The terms are 5𝑥, −3𝑦, 6𝑧, −1
The coefficients are 5, −3, 6 and the constant is −1

19. Terms, Like Terms, and the Distributive Property
The expression 5𝑥−3𝑦+6𝑧−1 cannot be written in a simpler form (unless values are known or given for the variables)
However, the expression 5𝑥+2+7𝑥+9 is different
*Can we rearrange the addition operations (think Property!)?
*Can we group them differently (Property?)

20. Terms, Like Terms, and the Distributive Property
We can use the Commutative Property for Addition to get 5𝑥+2+7𝑥+9=5𝑥+7𝑥+2+9
We can use the Associative Property for Addition to get 5𝑥+7𝑥 +(2+9)
Now, we know that 2+9=11; what about 5𝑥+7𝑥?
*Remember that the factoring form of the Distributive Property tells us that 𝑎𝑏+𝑎𝑐=𝑎(𝑏+𝑐)
*Notice that 5⋅𝑥+7⋅𝑥 has a common factor:_____

21. Terms, Like Terms, and the Distributive Property
*Now we have 5⋅𝑥+7⋅𝑥=𝑥⋅5+𝑥⋅7=𝑥 5+7 =𝑥⋅12=12𝑥
*When two or more terms of an expression have the same variable as a common factor, then we can call these like terms
*By the Distributive Property, we can show that it is always possible to add (subtract) the coefficients of like terms; we call this combining like terms

22. Terms, Like Terms, and the Distributive Property
Now we can solve equations with variables on both sides of the equal sign
Remember that our definition of solving a linear equations says that the variable is to be isolated on one side of the equation
*So when an equation has like terms on both sides, use the Inverse Property of Addition to eliminate the variable terms on one side of the equal sign
*Solve for 𝑥: 4𝑥−7=2𝑥+9

23. A Word About the Reciprocal of a Number
As you have seen, the Inverse Property of Multiplication states that, for every nonzero number 𝑎, there exists a nonzero number 1 𝑎 such that 𝑎⋅ 1 𝑎 = 1 𝑎 ⋅𝑎=1
What if 𝑎 is a fraction?
*In that case, the reciprocal of a number 𝑚 𝑛 is simply the number 𝑛 𝑚
*Solve the equation 2 3 𝑥−7=4

24. Guided Practice Find the solution set for each equation. 3−4𝑛=23
7𝑥−3=2𝑥+12
2 2𝑥+1 =3(𝑥−7)
−1= 𝑝

25. Guided Practice Find the solution set for each equation. 5= 1 3 𝑎−2
4 5 𝑦−4=16
8𝑛−3=6−𝑛
5𝑔+4=3(2𝑔+1)

26. Concentrate!