Algebra 2: Solving Equations - Properties of Equality

Algebra 2: Solving Equations - Properties of Equality

by Faith Windsor

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​Algebraic expressions lack one thing - an equal sign!

​BUT... we can connect TWO expressions together with an equal sign. This makes an equation!

​

​Think about an equal sign. Or just the word equal. I can't say that the my daily work-out equals. Equals what? It has to have an answer, a solution if you will. Now, if I say that my daily work-out equals 30 minutes, the statement actually takes on meaning.

​The purpose of an equation is to find that answer, that solution. In Algebra, this usually means finding what a variable equals, or what it solution is.

Multiple Choice

If a = a, Which of the following options is a true statement?

• x = 7

• 2x - 7 = 2x + 7

• 2x = 7

• 2x - 7 = 2x - 7

Fill in the Blank

If a = b, and b = c, then what can you conclude? What other terms equal each other?

Fill in the Blank

a = b, and b = a.

Therefore, if -3y = 6, then _____

Poll

If a = b, can I add 3 to both sides?

• yes

• no

Poll

If  $$-\frac{3}{5}y=9$$  Determine if the following statement is true or false.

$$-\frac{5}{3}\left(-\frac{3}{5}y\right)=-\frac{5}{3}\left(9\right)$$  

• true

• false

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​What you just reviewed are the Properties of Equality. You must have a strong understanding and application of these in order to solve equations effectively.

​

​The Addition & Multiplication POEs require that you perform the SAME operation on both sides of the equation.

Remember, that " - 8" is the same

as "+ (-8)".

​Remember, that " " is the same

as "       ".

​So How Do We Solve Equations?

​Well, we want to find the answer, the solution to our statement. Again, "my daily work-out equals" makes no sense without a measurable value. So how do I figure that out? Well we just reviewed all of our Properties of Equality, or POEs. We're going to use those to "undo" our algebraic equations using inverse operations. For example...

​x + 3 = 7 This reads " x plus 3 equals 7." What is the inverse of plus 3?

SUBTRACT 3! So let's do that! But by the Addition POE, we have to do

this on both sides

x + 3 = 7

-3 -3

​x = 4 Do we know what our variable equals? YES! x = 4. Now that we have a measurable solution, we have our final answer! Let's try some more.

​So how to solve equations...

​Solve 5x + 12 = -18

​5x + 12 = -18

​

​ -12 -12

​

​5x = -30

​ 5       5

​

​x = -6

• ​Rewrite the question.

• ​

• ​Subtract 12 on both sides using Addition/Subtraction POE.

• ​

• ​Divide both sides by 5 using Multiplication/Division POE.

• ​Final Answer

​___ ___

Solve -3(x + 2) = 4(x + 18) - 1

-3(x + 2) = 4(x + 18) - 1

​-3x + -3(2) = 4x +4(18) - 1

-3x + -6 = 4x + 72 -1

​-3x -6 = 4x + 71

+6 +6

​-3x = 4x + 77

​-4x   -4x

​-7x = 77

​-7 -7

​x = -11

• ​Rewrite the question.

• ​Distributive Property

• ​Simplify and Combine Like Terms.

• ​

• ​Add 6 to both sides using Addition POE.

• ​Subtract 4x from both sides of the equation using Addition POE.

• ​

• ​Divide by -7 on both side of the equation using the Multiplication/Division POE. Final Answer.

​___ ___

Solve

​

​

​

​​3x + 4(x-1) = 2(x + 3)

​3x + 4x +4(-1) = 2x + 2(3)

​3x + 4x -4 = 2x + 6

​7x - 4 = 2x + 6

​     +4          +4

​7x = 2x +10

​-2x  -2x

​5x = 10

​5       5 x = 2

• ​Find a common denominator for all three fractions. This is done using an LCD of 12.

• ​

• ​Simplify.

• ​Distributive Property

• ​Simplify using Multiplication. Combine Like Terms.

• ​Add 4 to both sides of eq.n using Add. POE.

• ​Subtract 2x from both sides of eq.n using Add./Subtr. POE.

• ​Divide by 5 on both sides of eqn. using Mult./Division POE.

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