Equivalent equations

EquationsIts all about equivalence

An equation ….

Is a mathematical sentence that says that two expressions are equal.

What is on the left = what is on the right

There are two types of equations …..

Numerical5 + 9 = 14

Algebraic2x – 3 = 14

Modeling of equations

2x + 4 = -6

Model these in your notes:

-3x + 4 = 2x

5+ 2x = -1

6x – 3 = -2x + 4

Our goal with equations is to:

• Find the value of “x” or the variable

• Solve it

To do this we must rewrite the equation in progressively simpler equivalent forms of the original

equation.

So to do this we need to use the most important rule for

equations …..

What you do to one side you MUST do to the other

2x + 4 = -6

From the equation we can choose to try the following:

+ 2x -2x

2x + 4 = -6

From the equation we can choose to try the following:

+ 2x -2x + 4 -4

2x + 4 = -6

From the equation we can choose to try the following:

+ 2x -2x + 4 -4 -6 + 6

2x + 4 = -6

From the equation we can choose to try the following:

+ 2x -2x + 4 -4 -6 + 6

x 2 or ÷2

2x + 4 = -6

+ 2x to both sides

4x + 4 = 2x - 6

- 2x on both sides

4=-2x-6

+ 4 on both sides

2x + 8 = -2

-4 on both sides

2x = -10

-6 on both sides

2X – 2 = -12

+6 on both sides

2x + 10 = 0

Multiply both sides by 2

4x + 8 = -12

Divide both sides by 2

Divide both sides by 2

x + 2 = -3

None of these steps are WRONG … an equivalent equation results every time

BUTthe ONE operation that makes the

equation simpler (shorter) is the one you are looking for!

For these, list all the possible things you could do to each side

-3x + 4 = x

5+ 2x = -1

5x – 3 = -2x + 4

-3x + 4 = x

-3x + 3x

+ 4 -4

+x -x

x -3 ÷ -3

5+ 2x = -1

+ 5 -5

+ 2x -2x

-1 +1

x 2 ÷ 2

5x – 3 = -2x + 4

+ 5x -5x

-3 +3

-2x +2x

+ 4 -4

x 5 ÷ 5

x -2 ÷ -2

For the each of the 3 equations:

- Write out the equivalent form for each

2x + 4 = -6 Possible Resulting equivalent equation

Model of original +2x 4x + 4 = 2x - 6

-2x 4 = -2x - 6

+4 2x + 8 = -2