Equations Courtesy of Les Edition CEC

Khan Academy: Equations Review Evaluating Expressions in One Variable Evaluating Expressions in 2 Variables Evaluating Expressions with Variables Word Problems Identifying Parts of Expressions Why we do the same thing to both sides: Simple Equations One Step Equation Intuition Intuition Why We Divide Both Sides Constructing and Solving Equations in the Real World 1 Find Out Missing Algebraic Step Understanding the Process for Solving Linear Equations Variables on Both Sides Equations with Variables on Both Sides Solving Equations in Terms of a Variable (advanced)

Equations Numbers, symbols and operators (such as + and ×) grouped together that show the value of something. Ex: 2x + 10 = 5

Equations vs. expressions An equation says that two “things” are equal. These two “things” that we are comparing are mathematical expressions. An equal sign (=) is always used. Ex: 6 + 2 = 10 – 2

Equations vs Expressions EQUATIONS (= sign) Compares two expressions EXPRESSIONS (no = sign) 4x - 8 = 4 3x2 - 2 142 - 28x = 12x + 73 3a + 4b + 12 7 + 2 = 10 - 1 10 + 2x 4a + b = 8 3 + 2

Equivalent Equations Equations are equivalent if they have the same solution(s) 2x = 14 5x = 3x + 14 5x - 8 = 3x + 6 All three of these equations are “equivalent equations” since x = 7 is the solution for each of them.

Equivalent equations Proof:

Solving an Equation The point is to get the variable alone (isolate) on one side of the equation. The “equals” sign acts as the centre of a balance. What you do on one side of the equation, you must do on the other side! ? + 3 = 10

Types of solutions There are three types of solutions for an equation. One solution x = 7 No solution when the variable cancels out 2 = x - x 2 ≠ 0 3) Infinite solutions x = x GRAPH

Solving an equation BEDMAS (if there is anything to reduce). Ex: 𝟑(𝟐𝒙+𝟔)=𝟐(𝟖 𝒙 −𝟒 )

Solving an equation 2) Put likes terms on one side of the equal sign. Do inverse operations! Reduce! 𝟔𝒙+𝟏𝟖=𝟏𝟔𝒙 −𝟖

Solving an equation 3) The x is attached to the -10 by a multiplication. You must do the equal and opposite of that, which means you divide by -10. −𝟏𝟎𝒙=−𝟐𝟔

Solving Equations - Example 1 What you do on one side of the equation, you must do on the other side!

Solving Equations Proof

Solving Equations - Example 2 Continued on next slide

Solving Equations - Example 2

Solving Equations Proof

Constructing an Expression or an Equation Word problems Ex: The sum of Claude and Jean’s ages is 52 years. Jean is 10 years older than twice Claude’s age. Determine their ages.

Constructing an Expression or an Equation Ex: The sum of Claude and Jean’s ages is 52 years. Jean is 10 years older than twice Claude’s age. Determine their ages. Step 1: Determine the unknown(s) The unknowns are: Claude’s age Jean’s age

Constructing an Expression or an Equation Ex: The sum of Claude and Jean’s ages is 52 years. Jean is 10 years older than twice Claude’s age. Determine their ages. Step 2: Represent each unknown with a variable or expression Let: x represent Claude’s age Jean’s age: 10 + 2x

Constructing an Expression or an Equation Ex: The sum of Claude and Jean’s ages is 52 years. Jean is 10 years older than twice Claude’s age. Determine their ages. Step 3: Construct an equation Claude’s Age + Jean’s Age = 52 x + (10 + 2x) = 52

Solving Equations Ex: The sum of Claude and Jean’s ages is 52 years. Jean is 10 years older than twice Claude’s age. Determine their ages.

Solving Equations Ex: The sum of Claude and Jean’s ages is 52 years. Jean is 10 years older than twice Claude’s age. Determine their ages. Since x = 14, we can now determine Jean’s age (PLUG 14 into x !!!) Jean’s Age = 10 + 2x = 10 + 2(14) = 38