Algebra 1 lessonplan powerpoint

Algebra 1

Solving Equations with Variables on BOTH sides


Variable: In algebra the variables are letters used to hold the place of a missing number in an equation. Most commonly used variables are ‘x’ and ‘y’Any letter may be used to hold the place of the missing number in the equation.


Variable: In algebra the variables are letters used to hold the place of a missing number in an equation. Most commonly used variables are ‘x’ and ‘y’Any letter may be used to hold the place of the missing number in the equation.

Our goal today: To solve an equation that has a variable on BOTH sides of the equation.

REVIEW: 6X+4=28

REVIEW: 6X+4=28Solve this equation: 6x+4=28

REVIEW: 6X+4=28Solve this equation: 6x+4=28 Subtract the 4 from both sides of the equation6x+4=28 -4 -4 6x =24

REVIEW: 6X+4=28Solve this equation: 6x+4=28 Subtract the 4 from both sides of the equation6x+4=28 -4 -4 6x =24Now divide each side by 66x = 24 6 6

We now have gotten ‘x’ by itself and solved the equation

x=4

Taking the Next Step::


What does an equation with 2 variables look like??


What does an equation with 2 variables look like??

Examples:

4x+13=12x+45 3x+8=2x+10

5x+7=9x-5 9x-17=3x-23

Step by Step we solve the equation:


Step 1: Combining like terms and rearranging the equation so we can easily solve the problem. First we move the numbers which have no variable attached to them.



Step 2: Move the numbers WITH variables attached to them



Step 2: Move the numbers WITH variables attached to them

Step 3: Divide each side of the equation by the number attached to the variable

Step 1: Moving the Number without the Variable


4x+13=12x+45 3x+8=2x+10

5x+7=9x-5 9x-17=3x-23


4x+13=12x+45 3x+8=2x+10

-13 -13 -8 -8

5x+7=9x-5 9x-17=3x-23

+5 +5 +23 +23


4x+13=12x+45 3x+8=2x+10

-13 -13 -8 -8

4x =12x+32 3x =2x+2

5x+7=9x-5 9x-

17=3x-23 +5 +5

+23 +235x+12=9x

9x+6=3x

Step 2: Moving the number WITH the variable


4x+13=12x+45 3x+8=2x+10 -13 -13 -8 -8 4x =12x+32 3x =2x+2-12x -12x -2x -2x _

5x+7=9x-5 9x-17=3x-23 +5 +5 +23 +235x+12=9x 9x+6=3x-5x -5x -9x -9x _


4x+13=12x+45 3x+8=2x+10 -13 -13 -8 -8 4x =12x+32 3x =2x+2-12x -12x -2x -2x _-8x = 32 1x = 2

5x+7=9x-5 9x-17=3x-23 +5 +5 +23 +235x+12=9x 9x+6=3x-5x -5x -9x -9x _ 12= 4x 6 = -6x

Step 3: Dividing each side by the number attached to the variable

4x+13=12x+45 3x+8=2x+10 -13 -13 -8 -8 4x =12x+32 3x =2x+2-12x -12x -2x -2x _-8x = 32 1x = 2-8 -8 1 1 5x+7=9x-5 9x-17=3x-23 +5 +5 +23 +235x+12=9x 9x+6=3x-5x -5x -9x -9x _ 12= 4x 6 = -6x 4 4 -6 -6

SOLUTIONS

4x+13=12x+45 -13 -134x =12x+32-12x -12x _-8x = 32-8 -8Solution x= -4

SOLUTIONS

4x+13=12x+45 3x+8=2x+10 -13 -13 -8 -8

4x =12x+32 3x =2x+2-12x -12x -2x -2x _-8x = 32 1x = 2-8 -8 1 1Solution x= -4 Solution x=2

SOLUTIONS

4x+13=12x+45 3x+8=2x+10 -13 -13 -8 -8

4x =12x+32 3x =2x+2-12x -12x -2x -2x _-8x = 32 1x = 2-8 -8 1 1Solution x= -4 Solution x=2 5x+7=9x-5 +5 +55x+12=9x -5x -5x 12= 4x 4 4Solution x=3

SOLUTIONS

4x+13=12x+45 3x+8=2x+10 -13 -13 -8 -8

4x =12x+32 3x =2x+2-12x -12x -2x -2x _-8x = 32 1x = 2-8 -8 1 1Solution x= -4 Solution x=2 5x+7=9x-5 9x-17=3x-23 +5 +5 +23 +235x+12=9x 9x+6=3x-5x -5x -9x -9x _ 12= 4x 6 = -6x 4 4 -6 -6Solution x=3 Solution x= -1

SOLUTIONS

Check your work!!

Check your work!!

Plug the solution into the original problem

Check your work!!


Let’s check our solution for example problem #1

Check your work!!



4x+13=12x+45 where x= -4

Check your work!!



4x+13=12x+45 where x= -4

4(-4)+13 = 12(-4)+45

Check your work!!



4x+13=12x+45 where x= -4

4(-4)+13 = 12(-4)+45 -16 + 13 = -48 + 45

Check your work!!



4x+13=12x+45 where x= -4

4(-4)+13 = 12(-4)+45 -16 + 13 = -48 + 45 -3 = -3

Checking our solution for example problem #2


3x+8=2x+10 where x=2



3(2)+8=2(2)+10



3(2)+8=2(2)+10 6 +8= 4 +10



3(2)+8=2(2)+10 6 +8= 4 +10 14 = 14


5x+7=9x-5 where x=3


5x+7=9x-5 where x=3

5(3)+7=9(3)-5


5x+7=9x-5 where x=3

5(3)+7=9(3)-5 15 +7= 27 -5


5x+7=9x-5 where x=3

5(3)+7=9(3)-5 15 +7= 27 -5 22 = 22


9x-17=3x-23 where x= -1


9x-17=3x-23 where x= -1

9(-1)-17=3(-1)-23


9x-17=3x-23 where x= -1

9(-1)-17=3(-1)-23 -9 -17= -3 -23


9x-17=3x-23 where x= -1

9(-1)-17=3(-1)-23 -9 -17= -3 -23 -26 = -26

Your turn

Your turn

Solve the equation:

3x+4=13x-6

Your turn

Solve the equation:

3x+4=13x-6

1.Rearrange the equation 10x=10x

Your turn

Solve the equation:

3x+4=13x-6


2.Divide by 10 x=1

Your turn

Solve the equation:

3x+4=13x-6


2.Divide by 10 x=1

3.Check your solution 3+4=13-6

Your turn

Solve the equation:

3x+4=13x-6


2.Divide by 10 x=1

3.Check your solution 3+4=13-6

7 = 7

Work for the problem 3x+4=13x-6


3x+4=13x-6 +6 +6


3x+4=13x-6 +6 +6 3x+10=13x


3x+4=13x-6 +6 +6 3x+10=13x-3x -3x


3x+4=13x-6 +6 +6 3x+10=13x-3x -3x 10=10x 10 10


3x+4=13x-6 +6 +6 3x+10=13x-3x -3x 10=10x

x=1 10 10

Solve: 15x+4=8x+32

Solve: 15x+4=8x+32

1. Rearrange equation 7x=28

Solve: 15x+4=8x+32

1. Rearrange equation 7x=282. Divide by 7 x=4

Solve: 15x+4=8x+32

1. Rearrange equation 7x=282. Divide by 7 x=43. Check Solution

15(4)+4=8(4)+32 60

+4= 32 +32

Solve: 15x+4=8x+32

1. Rearrange equation 7x=282. Divide by 7 x=43. Check Solution

15(4)+4=8(4)+32 60

+4= 32 +32 64

= 64

Work for the problem 15x+4=8x+32


15x+4=8x+32 -4 - 4 15x = 8x+28


15x+4=8x+32 -4 - 4 15x = 8x+28 -8x -8x _ 7x = 28


15x+4=8x+32 -4 - 4 15x = 8x+28 -8x -8x _ 7x = 28 7 7


15x+4=8x+32 -4 - 4 15x = 8x+28 -8x -8x _ 7x = 28 7 7

x = 4

Special Cases

Special Cases

There are 2 ‘special cases’ in solving these types of equations.

Special Cases


The first will leave you with NO SOLUTION where the second will leave you with an infinite amount of solutions.

Special Cases



If there is NO SOLUTION this means no matter what you plug in for the variable the equation will never work

Special Cases



If there is NO SOLUTION this means no matter what you plug in for the variable the equation will never work

If there is an infinite amount of solutions it means that ANY number you plug in for the variable will cause the equation to be equal and correct.

Special Case #1

Special Case #1NO SOLUTION


4x-2=4x+6-4x -4x _

-2 = 6


4x-2=4x+6-4x -4x _

-2 = 6This can NEVER be true.


4x-2=4x+6-4x -4x _

-2 = 6This can NEVER be true. Answer: NO SOLUTION

Special Case #2

Special Case #2Infinite number of

solutions


solutions7x+4=2x+5x+4

7x+4=7x+4-7x -7x _

4 = 4


solutions7x+4=2x+5x+4

7x+4=7x+4-7x -7x _

4 = 4

This is ALWAYS true. So ‘x’ can equal ANY number

Special Case #2

Special Case #2 Additional method of solving


7x+4=2x+5x+47x+4=7x+4

-4 -4_7x = 7x

7 7x=x


7x+4=2x+5x+47x+4=7x+4

-4 -4_7x = 7x

7 7x=x

Shows that ANY number can be the solution

What if the problem is set up differently?


Problems may be set up with additional steps that must be taken prior to being able to solve the equation. The main focus will be to know how to manipulate the equation so it is set up in the way we just learned how to solve it.


Problems may be set up with additional steps that must be taken prior to being able to solve the equation. The main focus will be to know how to manipulate the equation so it is set up in the way we just learned how to solve it.

Never fear a problem. SOLVE it!

3(2x-4)=4x+8

3(2x-4)=4x+8

Let’s Solve:

3(2x-4)=4x+8 Distribute the 3 to the 2x & -4

3(2x-4)=4x+8

Let’s Solve:


6x-12 =4x+8 Now the equation is ready to be solved.

3(2x-4)=4x+8

Let’s Solve:



-4x -4x _

2x-12= 8

3(2x-4)=4x+8

Let’s Solve:



-4x -4x _

2x-12= 8

+12 +12

2x = 20

3(2x-4)=4x+8

Let’s Solve:



-4x -4x _

2x-12= 8

+12 +12

2x = 20

2 2

3(2x-4)=4x+8

Let’s Solve:



-4x -4x _

2x-12= 8

+12 +12 x=10 2x = 20

2 2

Now you know…..

Algebra 1 lessonplan powerpoint

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