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Solving Linear |
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Stick Diagram Solutions: Second Example
(unknown on one side, whole number coefficients)See too the similar Animated Example below: Solving 3x + 4 = 10 instead of solving 3x + 10 = 32.
Now in the equation 3x + 10 = 32 we imagine that x represents an unknown length. In the stick below, the top stick has length 3x + 10 while the bottom stick has length 32. The equation say both sticks have the same length, here 32.
3x
10
32
Cutting off or subtracting 10 from both sticks (adding -10) gives
3x
22
a stick of length 3x on top and a stick of length 22 = 32 - 10 on bottom. This second stick diagram represents the equation 3x =22.
But 3x = x + x + x. So third of the length of 3x is x. To find x, we will take cut each stick into thirds (three equal pieces)
x
x
x
22
3
22
3
22
3
The stick diagram suggests that x = 22
3= 7 1
3
x
22
3
A Solution Table
If I was solving 3x + 10 = 32 in class, I would just fill in the table and skip the work before it. 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.
Solution Table for 3x + 10 = 32
Stick Diagram Operation Equation
3x
10
32
Initial Situation
Given3x + 10 = 32
3x
22
Subtract 10
(a.k.a Add -10)3x = 22
x
22
3
A third of 3x is x. So x is also a third of 22.
Multiply by
1
3
to go from 2nd row to 3rd
x = 22
3= 7 1
3Check: Does
x = 22
3= 7 1
3satisfies the equation 3x + 10 = 32
Left Hand Side
= 3x + 10 Right Hand Side
= 32 = 3* 22
3+10 = 22 + 10
= 32So both sides of the equation are equal. Therefore
x = 22
3= 7 1
3satisfies the equation 3x + 10 = 32
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.
A Similar Example: 3x + 4 = 10 (animated gif)
Example: 2x + 9 = 19 (Animated Gif File)
