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Solving Linear |
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Stick Diagram Solutions: Fourth Example
(unknown on one side, unit fraction coefficients)See too Animated Example below
Now in the equation (½)x + 8 = 24½ we imagine that x represents an unknown length. In the stick below, the top stick has length (½)x+8 while the bottom stick has length 24½. The equation say both sticks have the same length, here 24½
(½)x
8
24½
Cutting off or subtracting 8 from both sticks gives
(½)x
16½
a stick of length (½)x on top and a stick of length 17 = 25 - 8 on bottom. This second stick diagram represents the equation (½)x =17.
Now x = (½)x + (½)x and 16½ + 16½ = 33. So double or multiply by two to get the diagrams
(½)x
(½)x
and hence
x
16½
16½
33
These stick diagram suggest that x =33. See check below.
A summary of the operation follow.
If I was solving (½)x + 8 = 24½ in class, I would just fill in the table and skip the work above. Each table consists of a diagram in the left column, a description of what is done or given in the middle column, and the equivalent equations in the rightmost column. At the moment, you are required to draw the stick diagram in the solution of the equation. That is a crutch. Later on, only the equation column is required with a few words to explain the operations.
Stick Diagram Operation Equation
(½)x
8
24½
Initial Situation
Given(½)x + 8 = 24½
(½)x
16½
Subtract 8
(½)x = 16½
(½)x
(½)x
16½
16½
Duplicate (a.k.a.
Replicate twice,
Multiply by 2, the
reciprocal of ½(½)x + (½)x =
16½ +16½
x
33
Simplify x = 33 Remember what we do one stick in a pair, we must do do the other, to keep the lengths after the operation, the same. If two sticks have the same length, we cut 5 from and 8 from the other, the resulting pair of sticks will not the same the length. And if two sticks in a pair have the same length, and I remove one half of one and two-thirds of the other, the resulting pair will have different lengths.
Check:
Is x = 33 a solution of (½)x + 8 = 24½? Now
Left Hand Side
= (½)x + 8
= (½)33 + 8
= 16½ + 8
= 24½Right Hand Side
= 24½ So both sides of the equation are equal when x = 33.
Note: We can always check whether a number is a solution of an equation or not by computing the left and right sides of an equation for or at that number. If the two sides differ, the number is not a solution.
- If you are asked to show that a number satisfies an equation you do the check.
- If you are asked to find a solution to an equation algebraically you should show some work (besides trial and error) that leads to the solution. Then you should check the solution.
Solutions of equations can always be checked. So before you hand-in an answer, you can always check whether it is correct or not. And if it is not correct, you should say so if you do not have time to find the correct answer. Instructors want to see how you obtain the solution. If your arithmetic without a calculator is usually good, the odd error in your work is not as important as you showing that you have master an algebraic method for solving problems.
Example: (½)x + 4 = 8 (animated gif)
Example: 1
3x + 5 = 12 (animated gif)
