Identity and Inverse Properties

Identity Property of Addition The Identity Property of Addition states that for any number x, x + 0 = x = = =¾ + 0 = ¾

Identity Property of Multiplication The Identity Property of Multiplication states that for any number x, x (1) = x Remember the number 1 can be in ANY form.

The number 1 can be in ANY form. In this case 3/3 is the same as 1. same

Inverse Property of Addition The inverse property of addition states that for every number x, x + (-x) = 0 4 and -4 are considered opposites.

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What number can be added to 15 so that the result will be zero? -15 What number can be added to -22 so that the result will be zero? 22

Inverse Property of Multiplication The Inverse Property of Multiplication states for every non-zero number n, n (1/n) = 1 The non-zero part is important or else we would be dividing by zero and we CANNOT do that.

Properties of Equality In all of the following properties Let a, b, and c be real numbers

Properties of Equality Addition property: If a = b, then a + c = b + c Subtraction property: If a = b, then a - c = b – c Multiplication property: If a = b, then ca = cb Division property: If a = b, then for c ≠ 0

Addition Property This is the property that allows you to add the same number to both sides of an equation. STATEMENTREASON x = ygiven x + 3 = y + 3Addition property of equality

Subtraction Property This is the property that allows you to subtract the same number to both sides of an equation. STATEMENTREASON a = bgiven a - 2 = b - 2Subtraction property of equality

Multiplication Property STATEMENTREASON x = ygiven 3x = 3yMultiplication property of equality This is the property that allows you to multiply the same number to both sides of an equation.

Division Property STATEMENTREASON x = ygiven x/3 = y/3Division property of equality This is the property that allows you to divide the same number to both sides of an equation.

More Properties of Equality Reflexive Property: a = a Symmetric Property: If a = b, then b = a Transitive Property: If a = b, and b = c, then a = c

Substitution Property of Equality If a = b, then a may be substituted for b in any equation or expression. You have used this many times in algebra. STATEMENTREASON x = x = y given = ysubstitution property of equality

Solving One-Step Equations

Definitions Term: a number, variable or the product or quotient of a number and a variable. examples: 12 z 2w c 6

Terms are separated by addition (+) or subtraction (-) signs. 3a – ¾b + 7x – 4z + 52 How many Terms do you see? 5

Definitions Constant: a term that is a number. Coefficient: the number value in front of a variable in a term.

3x – 6y + 18 = 0 What are the coefficients? What is the constant? 3, -6 18

Solving One-Step Equations A one-step equation means you only have to perform 1 mathematical operation to solve it. You can add, subtract, multiply or divide to solve a one-step equation. The object is to have the variable by itself on one side of the equation.

Example 1: Solving an addition equation t + 7 = 21 To eliminate the 7 add its opposite to both sides of the equation. t + 7 = 21 t = t = 14 t + 0 =

Example 2: Solving a subtraction equation x – 6 = 40 To eliminate the 6 add its opposite to both sides of the equation. x – 6 = 40 x – = x = 46

Example 3: Solving a multiplication equation 8n = 32 To eliminate the 8 divide both sides of the equation by 8. Here we “undo” multiplication by doing the opposite – division. 8n = n = 4

Example 4: Solving a division equation To eliminate the 9 multiply both sides of the equation by 9. Here we “undo” division by doing the opposite – multiplication.

Identify operations Undo operations Balance equation Repeat steps Solve for variable Check solution

Identify Operations Fraction bar means division Minus sign means subtraction

Use Opposite Operations or “undo” Operations Addition is opposite of subtraction (addition undoes subtraction) Subtraction is opposite of addition (subtraction undoes addition) Multiplication is opposite of division (multiplication undoes division) Division is opposite of multiplication (division undoes multiplication)

Keep Equation Balanced What ever you do to one side of the equation you do to the other side of the equation.

Repeat these steps until the equation is solved.

7x + 15 = 85 7x +15 – 15 = x = 70 7 x = 10 Example:

When graphing the solution to a linear equation with one- variable on a number line you would put a dot (point) on the answer. x – 3 = -7 x – = x = -4 Graphing a Linear Equation