Properties of Equality

Properties are rules that allow you to balance, manipulate, and solve equations

Adding the same number to both sides of an equation does not change the equality of the equation.

If a = b, then a + c = b + c. Ex: x = y, so x + 2 = y + 2

Subtracting the same number to both sides of an equation does not change the equality of the equation.

If a = b, then a – c = b – c. Ex: x = y, so x – 4 = y – 4

Multiplying both sides of the equation by the same number, other than 0, does not change the equality of the equation.

If a = b, then ac = bc. Ex: x = y, so 3x = 3y

A number is equal to itself. (Think mirror)

a = a Ex: 4 = 4

If numbers are equal, they will still be equal if the order is changed (reversed).

If a = b, then b = a. Ex: x = 4, then 4 = x

If numbers are equal to the same number, then they are equal to each other.

If a = b and b = c, then a = c. Ex: If x = 8 and y = 8, then x = y

Changing the order of addition or multiplication does not matter

oAddition: a + b = b + a

Multiplication: a ∙ b = b ∙ a

The change in grouping of three or more terms/factors does not change their sum or product.

o Addition: a + (b + c) = (a + b) + c

Multiplication: a ∙ (b ∙ c) = (a ∙ b) ∙ c

The sum of any number and zero is always the original number

a + 0 = a Ex: 4 + 0 = 4

The product of any number and one is always the original number.

Multiplying by one does not change the original number.

a ∙ 1 = a Ex: 2 ∙ 1 = 2

The sum of a number and its inverse (or opposite) is equal to zero.

a + (-a) = 0 Ex: 2 + (-2) = 0

The product of any number and its reciprocal is equal to 1.

a b• =1

b a

4 5• =1

5 4

The product of any number and zero is always zero.

a ∙ 0 = 0 Ex: 298 ∙ 0 = 0

1. Make sure each equation is in slope-intercept form: y = mx + b.

2. Graph each equation on the same graph paper.

3. The point where the lines intersect is the solution. If they don’t intersect then there’s no solution.

4. Check your solution algebraically.

1. Graph to find the solution.

4

2

y x

y x

2. Graph to find the solution.

2 5

2 1

y x

y x

If the lines have the same y-intercept b, and the same slope m, then the system is consistent-dependent.

If the lines have the same slope m, but different y-intercepts b, the system is inconsistent.

If the lines have different slopes m, the system is consistent-independent.

1. Arrange the equations with like terms in columns

2. Multiply, if necessary, to create opposite coefficients for one variable.

3. Add the equations. 4. Substitute the value to solve for

the other variable.5. Check

2 2 8

2 2 4

x y

x y

4x + 3y = 162x – 3y = 8

1. One equation will have either x or y by itself, or can be solved for x or y easily.

2. Substitute the expression from Step 1 into the other equation and solve for the other variable.

3. Substitute the value from Step 2 into the equation from Step 1 and solve.

4. Your solution is the ordered pair formed by x & y.

5. Check the solution in each of the original equations.

Get in slope-intercept form Determine solid or dashed line Determine whether to shade above or

shade below the line (Test Points) If the test point is true, shade the half

plane containing it. If the test point is false, shade the half

plane that does NOT contain the point

Shade above

> ≥

Shade Below

< ≤

Solid Line

≤ ≥

Dashed Line

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