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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
