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Solving Linear |
Read them in order
Example 4: Solve 3x - 4 =-11 (animated presentation)
Observe Error Recovery & Advice in Solution
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Example 5: Solve x + 12 = 10x - 30
| Operation | Equation | ||||||||
| given | x + 12 = 10x - 30 | ||||||||
| add -x to both sides | 12 = 9x - 30 | ||||||||
| add 30 to both sides | 42 = 9x | ||||||||
| Switch sides | 9x = 42 | ||||||||
| Divide both sides by 9 (or multiply by 1/3) |
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Note:
- We could have added -x + 30 to both sides in one step, not two.
- We switched sides for cosmetic reasons. That is we prefer to write b =15 in place of 15 = b.
- The switch would not have been needed if we had started with the equation 10x - 30 = x + 12 instead of x + 12 = 10x - 30. If x is a solution of one, it is also a solution of the other. This switch would have put the bigger coefficient on the left hand side and result in 9x = 42.
| Operation | Equation | ||||||||
| given | x + 12 = 10x - 30. | ||||||||
| switch sides to get bigger coefficient for x on left. |
10x - 30 = x + 12 | ||||||||
| add -x to both sides | 9x - 30 = 12 | ||||||||
| add 30 to both sides | 9x = 42 | ||||||||
| Divide both sides by 9 (or multiply by 1/3) |
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You see the solution does not change.
Check: For
| x = | 4 | 2 3 |
we have
| LHS = x+ 12 |
RHS = 10x-30 |
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| = | 4 | 2 3 |
+ | 12 | |||||||||
| = | 10 | (4 | 2 3 |
) | -30 | ||||||||
| = | 16 | 2 3 |
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| = | 40 | + | 20 3 |
- | 30 | ||||||||
| = | 10 | + | 6 | 2 3 |
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| = | 16 | 2 3 |
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Observe the left hand side (LHS) has the same value as the right hand side (RHS). So the value
| x = | 4 | 2 3 |
works.
Example 6: Solve -3x + 4 = 10
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Example 7: Solve 6x + 4 = 4x + 14
Solution Steps:
| Operation | Equation |
| given | 6x + 4 = 4x + 14. |
| add -4x to both sides | 2x + 4 = 14 |
| add -4 to both sides | 2x=10 |
| multiply both sides by ½ | x = 5 |
We could have added -4x + -4 to both sides in one step instead of two.
Check: For x =5
| Left Hand Side | = 6 x + 4 = 6*5+4 = 30+4 = 34 |
Right Hand Side | = 4x + 14 = 4*5+14 = 20 +14 = 34 |
So for x =5, the Left Hand Side and Right Hand Side have the same value.