algebra 1 lessonplan powerpoint
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Algebra 1
Solving Equations with Variables on BOTH sides
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.
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.
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::
Taking the Next Step::
What does an equation with 2 variables look like??
Taking the Next Step::
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 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 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 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 3: Divide each side of the equation by the number attached to the variable
Step 1: Moving the Number without 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
Step 1: Moving the Number without the Variable
4x+13=12x+45 3x+8=2x+10
-13 -13 -8 -8
5x+7=9x-5 9x-17=3x-23
+5 +5 +23 +23
Step 1: Moving the Number without the Variable
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
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 _
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 _-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!!
Plug the solution into the original problem
Let’s check our solution for example problem #1
Check your work!!
Plug the solution into the original problem
Let’s check our solution for example problem #1
4x+13=12x+45 where x= -4
Check your work!!
Plug the solution into the original problem
Let’s check our solution for example problem #1
4x+13=12x+45 where x= -4
4(-4)+13 = 12(-4)+45
Check your work!!
Plug the solution into the original problem
Let’s check our solution for example problem #1
4x+13=12x+45 where x= -4
4(-4)+13 = 12(-4)+45 -16 + 13 = -48 + 45
Check your work!!
Plug the solution into the original problem
Let’s check our solution for example problem #1
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
Checking our solution for example problem #2
3x+8=2x+10 where x=2
Checking our solution for example problem #2
3x+8=2x+10 where x=2
3(2)+8=2(2)+10
Checking our solution for example problem #2
3x+8=2x+10 where x=2
3(2)+8=2(2)+10 6 +8= 4 +10
Checking our solution for example problem #2
3x+8=2x+10 where x=2
3(2)+8=2(2)+10 6 +8= 4 +10 14 = 14
Checking our solution for example problem #3
Checking our solution for example problem #3
5x+7=9x-5 where x=3
Checking our solution for example problem #3
5x+7=9x-5 where x=3
5(3)+7=9(3)-5
Checking our solution for example problem #3
5x+7=9x-5 where x=3
5(3)+7=9(3)-5 15 +7= 27 -5
Checking our solution for example problem #3
5x+7=9x-5 where x=3
5(3)+7=9(3)-5 15 +7= 27 -5 22 = 22
Checking our solution for example problem #4
Checking our solution for example problem #4
9x-17=3x-23 where x= -1
Checking our solution for example problem #4
9x-17=3x-23 where x= -1
9(-1)-17=3(-1)-23
Checking our solution for example problem #4
9x-17=3x-23 where x= -1
9(-1)-17=3(-1)-23 -9 -17= -3 -23
Checking our solution for example problem #4
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
1.Rearrange the equation 10x=10x
2.Divide by 10 x=1
Your turn
Solve the equation:
3x+4=13x-6
1.Rearrange the equation 10x=10x
2.Divide by 10 x=1
3.Check your solution 3+4=13-6
Your turn
Solve the equation:
3x+4=13x-6
1.Rearrange the equation 10x=10x
2.Divide by 10 x=1
3.Check your solution 3+4=13-6
7 = 7
Your turn
Solve the equation:
3x+4=13x-6
1.Rearrange the equation 10x=10x
2.Divide by 10 x=1
3.Check your solution 3+4=13-6
7 = 7
Work for the problem 3x+4=13x-6
Work for the problem 3x+4=13x-6
3x+4=13x-6 +6 +6
Work for the problem 3x+4=13x-6
3x+4=13x-6 +6 +6 3x+10=13x
Work for the problem 3x+4=13x-6
3x+4=13x-6 +6 +6 3x+10=13x-3x -3x
Work for the problem 3x+4=13x-6
3x+4=13x-6 +6 +6 3x+10=13x-3x -3x 10=10x 10 10
Work for the problem 3x+4=13x-6
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
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
Work for the problem 15x+4=8x+32
15x+4=8x+32 -4 - 4 15x = 8x+28
Work for the problem 15x+4=8x+32
15x+4=8x+32 -4 - 4 15x = 8x+28 -8x -8x _ 7x = 28
Work for the problem 15x+4=8x+32
15x+4=8x+32 -4 - 4 15x = 8x+28 -8x -8x _ 7x = 28 7 7
Work for the problem 15x+4=8x+32
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
There are 2 ‘special cases’ in solving these types of equations.
The first will leave you with NO SOLUTION where the second will leave you with an infinite amount of solutions.
Special Cases
There are 2 ‘special cases’ in solving these types of equations.
The first will leave you with NO SOLUTION where the second will leave you with an infinite amount of solutions.
If there is NO SOLUTION this means no matter what you plug in for the variable the equation will never work
Special Cases
There are 2 ‘special cases’ in solving these types of equations.
The first will leave you with NO SOLUTION where the second will leave you with an infinite amount of solutions.
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
Special Case #1NO SOLUTION
4x-2=4x+6-4x -4x _
-2 = 6
Special Case #1NO SOLUTION
4x-2=4x+6-4x -4x _
-2 = 6This can NEVER be true.
Special Case #1NO SOLUTION
4x-2=4x+6-4x -4x _
-2 = 6This can NEVER be true. Answer: NO SOLUTION
Special Case #2
Special Case #2Infinite number of
solutions
Special Case #2Infinite number of
solutions7x+4=2x+5x+4
7x+4=7x+4-7x -7x _
4 = 4
Special Case #2Infinite number of
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
Special Case #2 Additional method of solving
7x+4=2x+5x+47x+4=7x+4
-4 -4_7x = 7x
7 7x=x
Special Case #2 Additional method of solving
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?
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.
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.
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:
3(2x-4)=4x+8 Distribute the 3 to the 2x & -4
6x-12 =4x+8 Now the equation is ready to be solved.
3(2x-4)=4x+8
Let’s Solve:
3(2x-4)=4x+8 Distribute the 3 to the 2x & -4
6x-12 =4x+8 Now the equation is ready to be solved.
-4x -4x _
2x-12= 8
3(2x-4)=4x+8
Let’s Solve:
3(2x-4)=4x+8 Distribute the 3 to the 2x & -4
6x-12 =4x+8 Now the equation is ready to be solved.
-4x -4x _
2x-12= 8
+12 +12
2x = 20
3(2x-4)=4x+8
Let’s Solve:
3(2x-4)=4x+8 Distribute the 3 to the 2x & -4
6x-12 =4x+8 Now the equation is ready to be solved.
-4x -4x _
2x-12= 8
+12 +12
2x = 20
2 2
3(2x-4)=4x+8
Let’s Solve:
3(2x-4)=4x+8 Distribute the 3 to the 2x & -4
6x-12 =4x+8 Now the equation is ready to be solved.
-4x -4x _
2x-12= 8
+12 +12 x=10 2x = 20
2 2
Now you know…..