Solving Equations w/ Variables on Both Sides

Solving Equations w/ Variables on Both Sides

​thanks to Big Ideas Math!

​Steps

• ​Distribute to remove parenthesis

• ​Convert or eliminate fractions

• ​Combine like terms on the same side of the equation

• ​Move variables to one side (doesn't matter which but the smaller coefficient is best) using inverse of addition/subtraction

• ​Move constants to the other side of the variable using invers of addition/subtraction

• ​Isolate the variable by using the inverse of multiplication/division

Multiple Select

What is the first step in solving

-3x = 2x + 15

• Distribute the x

• Move the -3x using addition

• Move the 2x using subtraction

• Add 2x and 19

Multiple Choice

The best thing was to move the 2x since the only constant was already on the right side!  Now what do you get when you subtract 2x from both sides?

-3x = 2x + 15

-2x = -2x      

• 1x

• 5x

• -1x

• -5x

Multiple Choice

Since they were both a negative you ADD and KEEP the signs!

Now what do you do?

-5x = 15

• Add 5 to both sides

• Subtract 5 from both sides

• Multiply both sides by 5

• Divide both sides by 5

Multiple Choice

Since the last step is -5 TIMES x, you have to use the INVERSE and DIVIDE by -5

What is the final answer?

-5x = 15

/5   /5

• x = 5

• x = -5

• x = 75

• x = -75

Fill in the Blanks

Try this one on your own.

Just write the number of your answer with no spaces.

m - 4 = 2m

Fill in the Blanks

Try this one on your own.

Just write the number of your answer with no spaces.

3k - 2 = 7k + 2

Multiple Choice

What step would you do first to solve

2(n - 3) = 4n + 1

• Distribute the 2 to the n - 3

• Distribute the 2 to all the terms

• Subtract n from 4n

• Add 3 to + 1

Multiple Choice

Distribute the 2 to n - 3 to remove the ( ). What do you get?

2(n - 3) = 4n + 1

• 2n - 3

• 2n - 6

• 8n 

• 4n

Multiple Choice

Remember to distribute to ALL numbers in the ( ).

What is the next step?

2n - 6 = 4n + 1

• Add 2n to both sides

• Subtract 2n from both sides

• Combine the 2n and -6

• Combine the 4n and 1

Multiple Choice

Moving the variables to one side is the best move.

You can move either, but most people like to move the smaller coefficient!

2n - 6 = 4n + 1

-2n       -2n        

• -6 = 6n + 1

• -6 = 2n + 1

• 4n - 6 = 6n + 1

• 4n - 6 = 2n + 1

Multiple Choice

You should now have the variables on one side only...so move the constants to the other side! How do we do this?

- 6 = 2n + 1

• Add 1 to both sides

• Subtract 1 from both sides

• Add 6 to both sides

• Subtract 6 from both sides

Multiple Choice

Watch your signs when doing these adding/subtracting steps!

- 6 = 2n + 1

-1           -1

• -5 = 2n

• 5 = 2n

• -7 = 2n

• 7 = 2n

Multiple Choice

Same signs mean ADD and KEEP. Now what do we do?

- 7 = 2n

• Divide by -7

• Divide by 2

• Multiply by -7

• Multiply by 2

• A

Multiple Choice

You always divide by the coeffcient with the variable...that is the SECOND number you put in your calculator! What is our final answer?

7 = 2n

/2   /2  

• 7

• 2

• -3.5

• 3.5

Multiple Choice

Solve the equation

4x - 5x + 9 =5x - 9

• x = -3

• x = 9

• x = 3

• x = -9

Multiple Choice

8x - 8 = -4x + 16

• 8

• 16

• 24

• 2

Multiple Choice

24 = 2a - 6

• a=-15

• a=15

• a= 9

• a= -9

Multiple Choice

2 + (¹∕₄)a = 10

• a= 32

• a= 48

• a= 2

• a= -2

Multiple Choice

3 + ^a⁄₃ = 6

• a= 1

• a= 3

• a= 9

• a= 15

Multiple Choice

50 = 28.1 + 0.15a

• a= 150

• a= 146

• a= 6

• a= 155

​Equations with "special" solutions

• ​When you solve an equation and all the variables are eliminated you can NOT have one solution...but you have 2 other options

• ​If the numbers left over make a TRUE statement; ex; 2 = 2 you say INFINITELY MANY SOLUTIONS

• ​If the numbers left over make a FALSE statement; ex; 7 = -2 you say NO SOLUTIONS

Multiple Choice

Solve:

5x + 4 = 5x − 3

• x = 4

• x = -3

• infinitely many solutions

• no solutions

Multiple Choice

Solve:

2y − 3 = 2y − 3

• y = 0

• y = -3

• infinitely many solutions

• no solutions

Multiple Choice

Solve:

3(x − 2) = 3x − 6

• x = 6

• x = -6

• infinitely many solutions

• no solutions

Multiple Choice

Solve:

6(y + 4) = 6y + 10

• y = 10

• y = 24

• infinitely many solutions

• no solutions

Multiple Choice

What does it mean when an equation has one solution?

• the equation is always TRUE when any value is substituted in for the variable

• the equation will NEVER be true when any value is substituted in for the variable

• only ONE solution will make the equation true

Multiple Choice

What does it mean when an equation has infinitely many solutions?

• "To infinity and beyond"

• the equation is always TRUE when any value is substituted in for the variable

• the equation will NEVER be true when any value is substituted in for the variable

• only ONE solution will make the equation true

Multiple Choice

What is the solution to this equation?

• one solution
• no solution
• infinite solutions
• none of these

Multiple Choice

What is the solution to the equations?

• one solution
• no solution
• infinite solutions
• none of these