Review Solving Linear Equations

Review Solving Linear Equations

Let's review and practice the steps to solving equations

One-Step Equations

• These equations take only one step to finish!

• 1.) Identify what operation is being done in the original equation

• 2.) Use the inverse of that operation to "undo" and solve the equation

• *Always remember to check your answer!

Multiple Choice

What operation is happening in the equation 6x = 72?

• Addition

• Subtraction

• Multiplication

• Division

Multiple Choice

What is the inverse operation to solve the equation 6x = 72?

• Addition

• Subtraction

• Multiplication

• Division

Two-Step Equations

• These equations take two steps to finish

• Some two step equations follow these steps...

• 1.) Move the constant by addition or subtraction

• 2.) Divide or multiply to isolate the variable

• An example of this would be 3x + 2 = 11, first you would subtract the 2 then divide by 3

Two-Step Equations (cont'd)

• Other two-step equations may look like this...

•  $\frac{x\ +\ 2}{8}=\ 1$  

• In this equation, there are 2 terms in the numerator, so we want to multiply by 8 first first, then subtract 2 to isolate the variable

• No matter the form, you are working the equation backwards or in reverse order of operations

• *Always remember to check your answer!

Fractions!

• Sometimes the variable has a fraction with it like this...

•  $\frac{2}{3}x\ =\ 2$  

• We still need to isolate the variable, so we have to get rid of that fraction 

• We need to use the reciprocal! 

• And what we do on one side we have to do on the other so it should look like this:

•  $\left(\frac{3}{2}\right)\ \frac{2}{3}x\ =\ 2\ \left(\frac{3}{2}\right)$  

• The fractions on the left cancel each other out leaving you with x. The right is simplified to 3, so x = 3

Fractions (con't)

• Another way to "get rid of" fractions is to multiply EVERYTHING by the denominator.

• Example:  $\frac{2}{3}x\ +\ 1\ =\ 5\$  , if we multiply EVERYTHING by 3, we get 2x + 3 = 15

• This leaves us with no fractions and a nice two-step equation we can solve

Multiple Choice

What are the steps to solve the equation 5x + 19 = 104

• Multiply by 5, then add 19

• Add 19, then multiply by 5

• Subtract 19, then multiply by 5

• Subtract 19, then divide by 5

• Multiply by 5, then subtract 19

Multiple Choice

What is the reciprocal of

 $\frac{6}{7}$  

•  $\frac{7}{6}$  

•  $\frac{6}{7}$  

Multiple Choice

What is the solution of the equation

 $10=\ 4\ +\ \frac{1}{2}x$  

• 12

• 14

• 16

• 3

Multi-Step Equations

• These equations take more than 2 steps to solve

• This includes steps like distributing

Distributing

• Distributing is done to simplify an equation that has parentheses

• To distribute, you take the term outside the parentheses and multiply it by EVERYTHING inside the parentheses (pay attention to signs)

• Example: -2(5x - 4)----> -2(5x) -2(-4)----> -10x + 8

Multiple Choice

What is the mistake this person made when they distributed?

3(-4 + 8x)--> 3(4) + 3(8x) --> 12 + 24x

• Nothing

• They should have added 3 instead of multiplying by 3

• It should be a -3 instead of 3

• They forgot the negative sign when they multiplied by 4

Combining Like Terms (Same Side)

• When combining like terms on the same side of the equal sign we do NOT have to use inverse operations

• Simply combine terms together to "clean up" each side of the equation

• Example: 2x + 5 + 9x - 4 = 34

• 2x and 9x are like terms and can be combined to be 11x

• 5 and -4 are also like terms and can be combined to be 1

• This makes our simplified equation: 11x + 1 = 34

Combining Like Terms (Opposite Sides)

• When combining like terms on opposite sides of the equal sign, we are actually moving them so we have to use inverse operations on each side

• 1.) Decide which terms are like terms and if they are positive or negative

• 2.) Decide which terms you are going to move

• 3.) If your moving a negative term, you need to add on both sides. If you are moving a positive term, you should subtract on both sides

• 4.) Make sure constants are on one side and variables are on the other

Multiple Select

Given the equation below, select all moves that would correctly combine like terms.

-3x + 8 = 10x - 12

• Add 12 to both sides

• Subtract 8 from both sides

• Add 8 to both sides

• Add 3x to both sides

• Subtract 10x from both sides

Multi-Step Equations

• When solving multi-step equations you only have to distribute or combine like terms when it applies

• Some examples are:

• If an equation does not have parentheses, you will not need to distribute

• If an equation does not have like terms on the same side, you will not combine them on the same side

Multiple Choice

What would this equation look like after distributing?

5 + 3(2x - 4) = 2x + 23

• 5 + 6x - 12 = 2x + 23

• 5 + 5x - 1 = 2x + 23

• 5 + 6x + 12 = 2x + 23

Multiple Choice

What would this equation look like after combining like terms on the same side?

5 + 6x + 12 = 2x + 23

• 6x + 17 = 2x + 23

• 8x = 40

• 6x + 7 = 2x + 23

• 23 = 25

Multiple Choice

Which of these is NOT an acceptable next step when solving 6x + 17 = 2x + 23?

• Subtract 6x from both sides

• Add 17 to both sides

• Subtract 17 from both sides

• Subtract 2x from both sides

Open Ended

Find the error(s) in the following equation.

 $\frac{1}{2}\left(6x\ +\ 4\right)\ =\ x\ +\ 2$ 
 
 $3x\ +\ 4\ =\ x\ +\ 2$ 

  $2x\ +\ 4\ =\ 2$  

 $2x\ =\ 6$  

 $x\ =\ 3$