Algebra 1 > Notes > Yorkcounty Final > Unit 2 > Lesson 5 - Solving_linear_equations

Solving Linear Equations Lesson 5

One-Variable Equations 

One-variable equations are mathematical expressions with an equal sign ( = ) in the middle and only 1 variable.



Example: x + 4 = 10

Properties of Equations 

We can think of think on being on a see-saw as a way to express equations. Our goal is to keep our see-saw always balanced.

Equations 

When we are working with equations we must have the same amount on both sides to keep our equation balanced. We want to keep our seesaw balanced. 4+ 5

=

9

Addition Property of Equality 

If we add something to one side of our equation, we must add it to both sides.



If we do not our equation will not be 9 balanced = 4+5+2

Addition Property of Equality 

So if we add 2 to the left side we must add it to both sides. 4+5+2



=

9+2

Now we have 11 = 11 , a balanced equation.

Subtraction Property of Equality 

We just learned that if we add something to one side of the equation we must add it to both sides. The same rule applies if we subtract something. We must do it to both sides. (4 + 5) – 3 3

=

9-

Subtraction Property of Equality 

This produces (4 + 5 ) – 3 9–3 6

=

9–3

=

6

=

6

Multiplication Property of Equality 

If we multiply something to one side of our equation we must do it to both sides.

(4 + 5)2 (9)2

=

Multiplication Property of Equality 

This produces (4 + 5)2

=

(9)2

(9)2

=

18

=

18

18

Division Property of Equality 

If we divide by something on one side we must divide by that something on both sides. (4 + 5) 3

=

9 3

Division Property of Equality 

This produces:

(4 + 5) (9) = 3 3 (9) =3 3

3=3

Solving Equations 

Now we can use these properties to help us solve these 1-variable equations.



Let us try an easy one x + 4 = 10

Solving Equations 

x + 4 = 10 ; we can see that we need to add 6 to 4 to get 10, but what would we do if we did not know that? What we want to do to solve this is to get the variable all by itself.



So how can we get x by itself on the left side?

Solving Equations 

x + 4 = 10 

What is in our way? The ( + 4 ) is on the side with our x. How can we get rid of it?



What we need to do is subtract 4 ( - 4 ). We subtract 4 ( - 4 ) because that is the opposite of ( + 4).



x + 4 – 4 = 10 – 4 .. Remember if we do something to one side of the equation we must do that to both sides.



Solving Equations 

 



x + 4 – 4 = 10 – 4 x=6 Now we have x alone on the left and on the right side of the equation we have 6. So we now know that x = 6. We need to check our answer.

Checking Solutions 

x + 4 = 10 ; we found that x = 6.



Substitute x = 6 for all x’s in the equation



(6) + 4 = 10 10 = 10 ; this produces a true statement so x = 6 is correct.

Strategies for Solving 



 

1) First eliminate any parentheses with distribution. 2) Use addition and subtraction properties of equality first. 3) Combine like terms 4) Use multiplication and division properties next.

Example 

Solve 2x + 4 = 20

First we want to get rid of the ( + 4 ). How can we get rid of that?

Example 

2x + 4 = 20

We need to subtract ( - 4 ) from both sides. 2x + 4 – 4 = 20 – 4 This produces 2x = 16

Example 

2x = 16

Now we need to get the x by itself. What is keeping the x from being alone? The 2 that is being multiplied to it is keeping it from being alone. What do we need to do to get rid of that 2? We need to do the opposite of multiplication. So we need to divide by 2.

Example

2 x 16 = 2 2

x =8

So now we found that x = 8, now we need to check our answers.

Check our Solution 

Make sure that you always check your answer in your original equation.



2x + 4 = 20 , we found that x = 8.

2(8) + 4 = 20 16 + 4 = 20 20 = 20, so x = 8 is a correct solution.

Example: 





Let us try a problem with a variable on both sides of the equation. 3x - 10 = x – 4 What you want to do first is to get all of the variables on one side of the equation. You want to move the smaller x. Between 3x and x you want to get rid of the x.

Example: 3x - 10 = x – 4 We want to subtract x from both sides. 

3x - 10 – x = x – x – 4 This produces 2x - 10 = - 4

Example: 

2x – 10 = -4

Now we need to get the x alone. First we need to get rid of the ( - 10 ). We need to add 10 ( + 10 ) to both sides of the equation.

Example: 

2x – 10 = -4

2x – 10 + 10 = - 4 + 10 This produces 2x = 6 What do we need to do now to get x alone. We need to divide by 2 ( the opposite of multiplication)

Example:

2x 6 = 2 2

x=3 Lets check our solution…..

Checking our Solution 

3x – 10 = x – 4 ; we found x = 3 3(3) – 10 = (3) – 4 9 – 10 = -1 -1 = -1 , this produces a true statement so x = 3 is the solution.