Some algebra basics:
Mathematics has a reputation for being a difficult subject, which tends to put people off studying it beyond school level, but it is in fact a very straightforward and logical subject once you get used to things like using abstract letters (such as x, y, z, a, b, c) to represent numbers, functions, unknowns in algebra,etc. Moreover, its a very versatile subject, and one in which its very useful to have a qualification when the time comes to look for a job.
So lets begin right here by explaining how these letters of the alphabet may be used in algebra.
Letters from the beginning of the alphabet (a, b, c, d, etc) are the ones normally used in algebra as unknowns in simple equations, as letters to be manipulated in 'simplify'-type questions, and as coefficients in harder equations. Letters from the end of the alphabet (x, y, z) usually represent unknown quantities to be solved for. (These are often referred to as 'variables', as opposed to the coefficients, which are constants..)
Here are some examples of each type:-
1. Solve the equation 3x + 2 = 17.
Clearly, the 'x' here represents the number 5, because 3x = 17 - 2 = 15.
[If you don't quite see how to solve this equation, here is the explanation:-
Think of the equation as a pair of scales. We can add the same amount to, or subtract the same amount from, both sides without upsetting the 'balance', We could also, if necessary, multiply or divide both sides by the same number without upsetting things. So, given the equation 3x + 2 = 17, you can subtract 2 from both sides without upsetting the 'balance'. This leads to 3x + 2 - 2 = 17 - 2, which simplifies to 3x = 15. Now divide both sides by 3, and this gives x = 5.]
2. Simplify the expression 12 a² b³
4ab
In this expression the letters a and b don't stand for anything in particular. They are just entities to be cancelled out to show you understand how to deal with powers (indices) in algebra.
After cancelling out between the top (numerator) and bottom (denominator) of the fraction we end up with 3ab² as the answer. (12/4 = 3 ; a²/a = a ; b³/b = b². Note that you have to deal with each bit separately.)
A word here about 'powers' of numbers or letters. The power of a letter is the number of times that letter is multiplied by itself. So b x b x b x b would be written as b^4 (read as 'b to the power 4').
Similarly, 7 x 7 x 7 could be written as 7^3 (read as '7 to the power 3') or 7³ (read as '7 cubed')
Powers of the same letter (or number) can be simplified as follows:-
b² x b³ = b^5, because if you multiply two b's by another three b's you are effectively multiplying five b's together. Hence the final power will be 5.
[Note that the symbol ^ stands for 'raised to the power' in this work.]
Similarly in division. b^5 ÷ b^3 = b^(5-3) = b^2 or b².
[because (b x b x b x b x b) ÷ (b x b x b) = b x b = b², the three b's underneath cancelling out three of the b's of the five b's above]
Indices are thus added when powers of the same letter are multiplied, and subtracted when they are divided. In cases where we are dividing a smaller power by a greater one, we have to introduce a negative index.
E.g. b^3 ÷ b^5 = b^(3 - 5) = b^ -2, read as 'b to the power minus two' .
Note that you can't combine powers of different letters. An expression such as a²b³ can't be reduced to anything simpler.
Fractional indices:-
The fractional index ½ stands for 'square root', so that b^½ means 'the square root of b'.
This follows from the rule of adding indices, as b^½ x b^½ = b^ (½ + ½) = b^1 = b.
Similarly, b^ ¼ would mean 'the fourth root of b', as you'd need to multiply b^¼ by itself FOUR times to get just b^1, or b. [b^(¼ + ¼ + ¼ + ¼) = b^1]
As an example of this, 81^¼ = 3, because 3^4 = 3 x 3 x 3 x 3 = 81.
A harder example is 81^¾ , which is a combination of roots and powers. The power ¾ means the fourth root raised to the power 3 (or 'cubed'), so that 81^¾ = 3³ = 27.
( N.B. 3³ is the same as 3^3)
Powers of letters and their manipulation are an important part of maths at all levels.
3. The coefficients of an equation are the numbers in front of the unknown letters x, y, etc.
Thus in the equation 5x + 3y = 9, the coefficients will be the numbers 5 and 3.
(Those of you who've done graphs may spot this as the equation of a straight line. Where does it cut the x-axis? Well, put y = 0 for this and you get x = 9/5, or 1.8. Similarly, it cuts the y-axis where x = 0, leading to 3y = 9, so y = 3)
More about equations:-
The simplest kind of equation contains just one unknown in it, as in 3x + 2 = 17 above, where x is the unknown and has to be found.
In the equation 5x + 3y = 9 there are two unknowns, the letters x and y. There's no single solution to this equation, because whatever value you give to x you'll always be able to find a corresponding value of y to fit this equation.
[E.g. suppose we give x the value 1, then 5x = 5, so in the equation we get 5 + 3y = 9. Subtracting 5 from both sides leads to 3y = 4, and now dividing both sides by 3 gives y = 4/3, or 1.33 (to 2 dec.pl.)]
Graphs:-
Graphs are drawn by first marking out two lines, one horizontally and one vertically, on the paper.
The horizontal one is called the x-axis, the vertical one is the y-axis. Each axis is then marked with a numbered scale, and to draw the graph of any given equation we mark a series of points corresponding to pairs of numbers, x and y, which fit the equation. This process is called 'plotting'.
[How to plot points:- Suppose we take the pair of numbers x = 1 and y = 1.33 which we found above to fit the equation 5x + 3y = 9. To plot these, go 1 unit along the horizontal (x) axis, then up 1.33 units parallel to the vertical (y) axis, and mark a small cross at the point reached. Do the same for the other pairs of numbers which fit the equation, which are best arranged in a 'table' of corresponding values.]
For simple equations where the graph is a straight line, just 3 points are enough (in fact, two would do, but the third one acts as a check). Such equations are called 'linear'.
But for harder equations such as y = x² - 5x + 6 (an example of a quadratic equation) the graph will be curved in shape, so many more points will need to be plotted for an accurate graph to be drawn.
Simultaneous equations:-
This term is used when we are given TWO linear equations to solve as a pair. In this case there will be just one answer - a unique pair of values for x and y which 'fit' BOTH equations (i.e. 'simultaneously').
Here's an example:-
Solve the simultaneous equations 2x + y = 3
3x - y = 7
Solution:- Add the two equations together. The +y and the -y will cancel out, leaving 5x = 10, or x = 2.
Now, to find y, substitute 2 for x in either equation.
So, if we choose the top one, 2x + y = 3, and put x = 2, we get 4 + y = 3, and subtracting 4
from both sides gives y = -1.
The answer is therefore x = 2, y = -1.
This example was deliberately made easier by having +y in one equation and -y in the other , so they cancelled out when added. But if one equation contained 2y and the other one -y, all terms in the second one would need to be multiplied by 2 so that the -2y would cancel the +2y when added.
Note that it needn't be the y terms which cancel. The x terms can be cancelled as well.
Here's another example to illustrate the method:-
Solve the simultaneous equations 3x + 2y = 3
2x + 3y = 7
Neither the x nor the y terms will cancel out as they stand, as they both have different coefficients.
Lets make the x-terms cancel this time.
So a way must first be found to make the number of x's the same in both equations. Then, if we subtract the equations, those letters will cancel out and we can solve for the remaining y's.
This is done by multiplying all terms of the first equation by 2 and all terms of the second by 3.
The equations now become 6x + 4y = 6
and 6x + 9y = 21
Now both equations contain the term 6x, and these will cancel out if subtracted. Taking the top equation from the bottom one then leads to 5y = 15, so y = 3. The answer for x is then found to be -1 (by substitution in either equation.)
(Note here that subtraction of an equation can be done by multiplying all its terms by -1 and then adding)
Solution by graphs:-
The same problem could also have been solved by drawing the graphs of the two given equations and noticing where they meet, or intersect. Each graph will be a straight line, and they should cross at the point where x = 2 and y = -1.
An example from Higher maths.
In an equation such as ax + by + cz = d the letters a, b, and c will be the coefficients.
Equations of this kind, with 3 unknowns, may actually represent the equation of a plane in 3-dimensions.
A numerical example would be 2x + 3y + 5z = 30.
I don't want to jump too far ahead here, as this topic is more likely to be found in the syllabus for A-level maths, yet its not too hard to see that by putting x = 0 and y = 0 you get 5z = 30, so z = 6. This means that, if you imagine 3 mutually-perpendicular axes in space, like the 3 edges at the corner of a box, with the z-axis vertical and the other two axes horizontal, the infinite plane represented by our equation will cut the z-axis at z = 6.
In the same way, you can find where this plane would cut the x- and y-axes as well. (Answer given below)
[Answer ; x = 15; y = 10 ]
Well, that completes this introductory section. Just for interest, though, letters from the Greek alphabet are much used in Mathematics, especially at the more advanced levels. What most people don't realise is that you can get the whole Greek alphabet on your computer if you go to Fonts, scroll down, and click on the one called 'Symbol'. Then, as you type out the English alphabet, you'll see the Greek letters appearing instead. You can also get these, and many other special mathematical letters and symbols, on your computer by going to START/All programs/accessories/system tools/character map.
Mathematics has a reputation for being a difficult subject, which tends to put people off studying it beyond school level, but it is in fact a very straightforward and logical subject once you get used to things like using abstract letters (such as x, y, z, a, b, c) to represent numbers, functions, unknowns in algebra,etc. Moreover, its a very versatile subject, and one in which its very useful to have a qualification when the time comes to look for a job.
So lets begin right here by explaining how these letters of the alphabet may be used in algebra.
Letters from the beginning of the alphabet (a, b, c, d, etc) are the ones normally used in algebra as unknowns in simple equations, as letters to be manipulated in 'simplify'-type questions, and as coefficients in harder equations. Letters from the end of the alphabet (x, y, z) usually represent unknown quantities to be solved for. (These are often referred to as 'variables', as opposed to the coefficients, which are constants..)
Here are some examples of each type:-
1. Solve the equation 3x + 2 = 17.
Clearly, the 'x' here represents the number 5, because 3x = 17 - 2 = 15.
[If you don't quite see how to solve this equation, here is the explanation:-
Think of the equation as a pair of scales. We can add the same amount to, or subtract the same amount from, both sides without upsetting the 'balance', We could also, if necessary, multiply or divide both sides by the same number without upsetting things. So, given the equation 3x + 2 = 17, you can subtract 2 from both sides without upsetting the 'balance'. This leads to 3x + 2 - 2 = 17 - 2, which simplifies to 3x = 15. Now divide both sides by 3, and this gives x = 5.]
2. Simplify the expression 12 a² b³
4ab
In this expression the letters a and b don't stand for anything in particular. They are just entities to be cancelled out to show you understand how to deal with powers (indices) in algebra.
After cancelling out between the top (numerator) and bottom (denominator) of the fraction we end up with 3ab² as the answer. (12/4 = 3 ; a²/a = a ; b³/b = b². Note that you have to deal with each bit separately.)
A word here about 'powers' of numbers or letters. The power of a letter is the number of times that letter is multiplied by itself. So b x b x b x b would be written as b^4 (read as 'b to the power 4').
Similarly, 7 x 7 x 7 could be written as 7^3 (read as '7 to the power 3') or 7³ (read as '7 cubed')
Powers of the same letter (or number) can be simplified as follows:-
b² x b³ = b^5, because if you multiply two b's by another three b's you are effectively multiplying five b's together. Hence the final power will be 5.
[Note that the symbol ^ stands for 'raised to the power' in this work.]
Similarly in division. b^5 ÷ b^3 = b^(5-3) = b^2 or b².
[because (b x b x b x b x b) ÷ (b x b x b) = b x b = b², the three b's underneath cancelling out three of the b's of the five b's above]
Indices are thus added when powers of the same letter are multiplied, and subtracted when they are divided. In cases where we are dividing a smaller power by a greater one, we have to introduce a negative index.
E.g. b^3 ÷ b^5 = b^(3 - 5) = b^ -2, read as 'b to the power minus two' .
Note that you can't combine powers of different letters. An expression such as a²b³ can't be reduced to anything simpler.
Fractional indices:-
The fractional index ½ stands for 'square root', so that b^½ means 'the square root of b'.
This follows from the rule of adding indices, as b^½ x b^½ = b^ (½ + ½) = b^1 = b.
Similarly, b^ ¼ would mean 'the fourth root of b', as you'd need to multiply b^¼ by itself FOUR times to get just b^1, or b. [b^(¼ + ¼ + ¼ + ¼) = b^1]
As an example of this, 81^¼ = 3, because 3^4 = 3 x 3 x 3 x 3 = 81.
A harder example is 81^¾ , which is a combination of roots and powers. The power ¾ means the fourth root raised to the power 3 (or 'cubed'), so that 81^¾ = 3³ = 27.
( N.B. 3³ is the same as 3^3)
Powers of letters and their manipulation are an important part of maths at all levels.
3. The coefficients of an equation are the numbers in front of the unknown letters x, y, etc.
Thus in the equation 5x + 3y = 9, the coefficients will be the numbers 5 and 3.
(Those of you who've done graphs may spot this as the equation of a straight line. Where does it cut the x-axis? Well, put y = 0 for this and you get x = 9/5, or 1.8. Similarly, it cuts the y-axis where x = 0, leading to 3y = 9, so y = 3)
More about equations:-
The simplest kind of equation contains just one unknown in it, as in 3x + 2 = 17 above, where x is the unknown and has to be found.
In the equation 5x + 3y = 9 there are two unknowns, the letters x and y. There's no single solution to this equation, because whatever value you give to x you'll always be able to find a corresponding value of y to fit this equation.
[E.g. suppose we give x the value 1, then 5x = 5, so in the equation we get 5 + 3y = 9. Subtracting 5 from both sides leads to 3y = 4, and now dividing both sides by 3 gives y = 4/3, or 1.33 (to 2 dec.pl.)]
Graphs:-
Graphs are drawn by first marking out two lines, one horizontally and one vertically, on the paper.
The horizontal one is called the x-axis, the vertical one is the y-axis. Each axis is then marked with a numbered scale, and to draw the graph of any given equation we mark a series of points corresponding to pairs of numbers, x and y, which fit the equation. This process is called 'plotting'.
[How to plot points:- Suppose we take the pair of numbers x = 1 and y = 1.33 which we found above to fit the equation 5x + 3y = 9. To plot these, go 1 unit along the horizontal (x) axis, then up 1.33 units parallel to the vertical (y) axis, and mark a small cross at the point reached. Do the same for the other pairs of numbers which fit the equation, which are best arranged in a 'table' of corresponding values.]
For simple equations where the graph is a straight line, just 3 points are enough (in fact, two would do, but the third one acts as a check). Such equations are called 'linear'.
But for harder equations such as y = x² - 5x + 6 (an example of a quadratic equation) the graph will be curved in shape, so many more points will need to be plotted for an accurate graph to be drawn.
Simultaneous equations:-
This term is used when we are given TWO linear equations to solve as a pair. In this case there will be just one answer - a unique pair of values for x and y which 'fit' BOTH equations (i.e. 'simultaneously').
Here's an example:-
Solve the simultaneous equations 2x + y = 3
3x - y = 7
Solution:- Add the two equations together. The +y and the -y will cancel out, leaving 5x = 10, or x = 2.
Now, to find y, substitute 2 for x in either equation.
So, if we choose the top one, 2x + y = 3, and put x = 2, we get 4 + y = 3, and subtracting 4
from both sides gives y = -1.
The answer is therefore x = 2, y = -1.
This example was deliberately made easier by having +y in one equation and -y in the other , so they cancelled out when added. But if one equation contained 2y and the other one -y, all terms in the second one would need to be multiplied by 2 so that the -2y would cancel the +2y when added.
Note that it needn't be the y terms which cancel. The x terms can be cancelled as well.
Here's another example to illustrate the method:-
Solve the simultaneous equations 3x + 2y = 3
2x + 3y = 7
Neither the x nor the y terms will cancel out as they stand, as they both have different coefficients.
Lets make the x-terms cancel this time.
So a way must first be found to make the number of x's the same in both equations. Then, if we subtract the equations, those letters will cancel out and we can solve for the remaining y's.
This is done by multiplying all terms of the first equation by 2 and all terms of the second by 3.
The equations now become 6x + 4y = 6
and 6x + 9y = 21
Now both equations contain the term 6x, and these will cancel out if subtracted. Taking the top equation from the bottom one then leads to 5y = 15, so y = 3. The answer for x is then found to be -1 (by substitution in either equation.)
(Note here that subtraction of an equation can be done by multiplying all its terms by -1 and then adding)
Solution by graphs:-
The same problem could also have been solved by drawing the graphs of the two given equations and noticing where they meet, or intersect. Each graph will be a straight line, and they should cross at the point where x = 2 and y = -1.
An example from Higher maths.
In an equation such as ax + by + cz = d the letters a, b, and c will be the coefficients.
Equations of this kind, with 3 unknowns, may actually represent the equation of a plane in 3-dimensions.
A numerical example would be 2x + 3y + 5z = 30.
I don't want to jump too far ahead here, as this topic is more likely to be found in the syllabus for A-level maths, yet its not too hard to see that by putting x = 0 and y = 0 you get 5z = 30, so z = 6. This means that, if you imagine 3 mutually-perpendicular axes in space, like the 3 edges at the corner of a box, with the z-axis vertical and the other two axes horizontal, the infinite plane represented by our equation will cut the z-axis at z = 6.
In the same way, you can find where this plane would cut the x- and y-axes as well. (Answer given below)
[Answer ; x = 15; y = 10 ]
Well, that completes this introductory section. Just for interest, though, letters from the Greek alphabet are much used in Mathematics, especially at the more advanced levels. What most people don't realise is that you can get the whole Greek alphabet on your computer if you go to Fonts, scroll down, and click on the one called 'Symbol'. Then, as you type out the English alphabet, you'll see the Greek letters appearing instead. You can also get these, and many other special mathematical letters and symbols, on your computer by going to START/All programs/accessories/system tools/character map.
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