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What does this scene remind you of? To most people, it is "balance." The understanding of balance is very critical when working with mathematical equations. |
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What does this scene remind you of? To most people, it is "balance." The understanding of balance is very critical when working with mathematical equations. |
Equations and Solving an Equation
The symbol to the right of the equal sign, 4, represents "four," while the symbols to the left, "1 + 3", represent "one plus three," which comes out to "four" also. Whenever you have an equal sign with symbols on both sides that represent the same quantity, you have an equation. As shown, we have been working with equations since kindergarten. |
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Of course, both sides of the equal sign must represent the same quantity in order to make up an equation. Therefore, the expression "2 + 1 = 4" is not a true equation; it is a false or wrong equation. In mathematics, we only work with true equations. |
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An equation can also have a literal symbol(s) in it, as is the case with
As explained in the section Variables, x represents an unknown and is called a variable. Since we only work with true equations, x can only represent those numbers that make a true equation out of the given expression. In the equation x + 3 = 4, it is easy to see that x must be equal to 1 in order to make the expression true. Why do we need to introduce this concept of equation? Some statements, either verbal or written, are too long or too complicated for our human brains to follow. When these statements are translated into algebraic forms, they become much shorter, thus easier to deal with. The following example explains why this is so. |
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You may wonder what type of rectangle this is and what the ratio of length to width might be. When starting to search for an answer, you may also find out that the statement in ordinary speech is a bit too long and/or too complicated to follow. However, this verbal statement can be easily written in algebraic form:
in which l represents the length and w represents the width. To further simplify the algebraic form (remember we want to find out the ratio?), we use a symbol ø (Greek letter phi) to represent the ratio of the length and the width "l / w". Thus, the above proportion will become the following equation:
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Those rectangles, whose length and width ratios satisfy the above equation, are called golden rectangles. This theory can be proved by the countless designs found all over the world. If you find certain picture frames, book covers, or buildings are pleasing to the eye, most likely the ratio of their length and width is very close to this golden ratio.
As shown, the algebraic form is much shorter. This is the power of algebra: results may be obtained without it, but they are more easily achieved with it. Now, the question is "How can we get this ratio from this equation?" |
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One way is the trial-and-error approach. Just try some numbers starting with zero, and increasing to larger numbers, and see what happens:
As the table shows, the correct answer should be between 1.5 and 1.7. Right away, we notice that this trial-and-error approach is very time-consuming. In addition, it is not reliable because it is almost impossible for us to find the right starting number at first. What happens if we decided to start with the number 2.0? We would have missed the correct answer. We cannot rely on such a method. The right way is to use the techniques that we will be learning in the next few sections, to solve the above equation. By solving an equation, we mean to find a proper number for x that makes the given expression true. |
[Experiment] [Exercise]