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Introduction to Algebra
What is Algebra?
Algebra consists of certain mathematical structures and methods through which variables and other values can be manipulated. Algebraic methods are used to maintain the mathematical integrity of equations while transforming them. Within equations and expressions, variables like x and y (See page on Variables and Functions) are used in place of unknown values. The algebraic method of finding the missing values of variables in equations often involves isolating variables and comparing them to known values. There are many forms of algebra, and it is used in many real life applications, including in finances, engineering, measurement, architecture and design.
Four Examples of Algebra
Example 1:
Find the value of x.
$$x\; + \;6\; = \;8$$
To answer this question, we isolate x by moving the 6 to the other side of the equation, while turning the plus into the inverse operation of minus as shown,
$$x\; = \;8\;-\;6$$
From this we can calculate the solution,
$$x\; = \;2$$
To check that this answer is correct, 2 is substituted for x in the original equation,
$$2\; + \;6\; = \;8$$
Since that is true, our answer of x = 2 is correct.
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$$x\;=\;14\;-\;9\;$$$$x\;=\;5\;$$
Example 2:
Find the value of x.
$$x\; – \;12\; = \;5$$
To answer this question, we isolate x by moving the minus 12 to the other side of the equation, while turning the minus into the inverse operation of plus as shown here,
$$x\;= \;12\; +\;5$$
From this we can calculate the following solution,
$$x\; = \;17$$
To check that this answer is correct, we substitute 17 into x in the original equation as follows,
$$17\;- \;12\; = \;5$$
Since that is true, our answer of x = 17 is correct.
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$$x\;=\;10\;+\;11\;$$$$\;x\;= \;21\;$$
Example 3:
Swapping Multiplication and Division
Find the value of x.
Phil rode around the bicycle track four and a half times before stopping. He noticed that his bike gauge said he had cycled a total of 2000 metres on the bicycle track. After he had arrived home he wanted to try to work out the distance of one lap of the cycle track. To find the answer to this question he made the following equation.
$$4 \frac{1} {2}x\;=2000\;$$
In the above equation, x stands for one lap of the cycle track. The next logical step would be to isolate x to make it easier to find its value. Think about how you could do that. We can isolate x by moving the four and a half to the other side of the equation and turning the multiplication that was between four and a half and x into the inverse operation of division as follows,
$$x\;=2000\;\div\;4 \frac{1} {2}$$
After that step, the final answer is easy to find. Remember to always state your answer in the unit you are dealing with. In this case it is metres.
$$x\;=444.44 \;metres$$
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$$x\;=7500\;\div\;7 \frac{1}{2} \;$$$$x\;=1000 \;metres$$
Example 4:
Find the value of x here, and also the inner angles of the triangle.
Study the diagram below,
We know that the sum of the interior angles of a triangle is always 180 degrees.
After observing the above diagram we can form the following equation,
$$19x\;+\;10x\;+7x\;=\;180$$
We know x will always be the same value, and thus can make the following addition by grouping together all like terms,
$$36x\;=\;180$$
The next logical step is to isolate x to make it easier to find its value. If we move the 36 to the other side of the equation and turn the multiplication that was between 36 and x into the inverse operation of division, then we can find the value of x.
$$x\;=180 \div 36\;$$
Then we state our solution,
$$x\;=\;5°$$
Now that we have that solution, we can easily find all our angles by multiplying them by the original figures in the inner angles of the triangle above, and they should all add up to 180 degrees.
$$19x\;=\;95°,\;10x\;=\;50° ,\;and\;7x\;=\;35°$$
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$$18x\;=\;180$$$$x\;=180 \div 18\;$$$$x\;=\;10°$$
More about the history of Algebra
Algebra was used in ancient times by classical Greek and Indian mathematicians to solve problems to do with geometry. Algebra comes from the Arabic word al-jabr, which is from a treatise whose title is translated as, The Compendious Book on Calculation by Completion and Balancing, written in 830 by the medieval Persian mathematician, Al-Khwarizmi. There have been many discoveries and developments in algebra and its uses since that time.
Four more algebra questions to challenge you
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$$x\;=\;17\;-\;6\;$$$$x\;=\;11\;$$
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$$x\;=\;98\;+\;5\;$$$$x\;= \;103\;$$
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$$42x\;=126\;$$$$x\;=126\;\div\;42 \;$$$$x\;=3 \;metres$$
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$$45x\;=\;180$$$$x\;=\;180\; \div\;45$$$$x\;= 4°$$
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