Sect. 1.3 Solving Equations  Equivalent Equations  Addition & Multiplication Principles  Combining Like Terms  Types of Equations  But first: Awards, HW Review, and Play ? 11.3

But, Before we go on… Let’s play Name That Law! a) x y = x + y + 5  Commutative Addition … COM + b) 3a + 6 = 3(a + 2)  Distributive … DIST c) 7x(1 / x) = 7  Reciprocals Multiplication … RECIP x d) (x + 5) + y = x + (5 + y)  Associative Addition … ASSOC + e) (y – 2)(3x)(y + 2) = (y – 2)(y + 2)(3x)  Commutative Multiplication … COM x f) 4(a + 2b) = 8b + 4a  COM, then DIST or DIST, then COM 21.3

Checking for Equivalent Equations A Solution is a Replacement Value that makes an equation True 31.3

Two Keys for Solving an Equation in One Variable  We need better techniques than guessing solutions  If we Add the same number to both sides of an equation, it will still have the original solution  If we Multiply both sides of an equation by the same non-0 number, it will still have the original solution 41.3

An Example of using Horizontal Technique for Applying the Addition Principle 51.3

Using the Vertical Technique to Solve the Same Equation 61.3

Using the Vertical Technique can Save Steps in more complex equations 71.3

An Example of Applying the Multiplication Principle 81.3

Combining Like Terms  A Term is the product of a coefficient and variable(s). Examples: 9x -2x 2 y 11 -p  Like Terms have identical variable parts Combine by adding their coefficients Example: 3xy + 11xy = (3+11)xy = 14xy Example: -2p + 6p – p = (-2+6-1)p = 3p Example: 4xyz + 6xy can’t be combined 91.3

Combining a Simple Expression 101.3

Simplify An Expression 111.3

The Opposite of an Expression 121.3

Another “Negative” Example 131.3

Connecting Concepts: Equations vs. Expressions 141.3

Solving Using Both Principles ( First the Addition Principle, then Multiplication) 151.3

 We have been solving Linear Equations  A linear equation is one that can be reduced to ax = b (a ≠ 0 and x is any variable to the 1 st power) Don’t rush to solve them in your head Work neatly, making each step result in an equivalent equation  Every linear equation will be in one of 3 categories: Conditional – it has only one value as a solution (2x = 4) Contradiction – no value will be a solution (x = x + 1) Identity – every value will make the equation true (x = x) Types of Linear Equations Identity – Contradiction - Conditional 161.3

Solve: Is it an identity, contradiction, or a conditional equation? 171.3

What’s Next?  1.4 Introduction to Problem Solving