Notesale: Turn your study into money

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How to solve linear equations
Properties of equalities
Two equations that have the same solution are called equivalent equations e
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5 +3
= 2 + 6
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An inverse operation are two operations that undo each other e
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addition and
subtraction or multiplication and division
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5+3−2=6+2−2
This gives us a couple of properties that hold true for all equations
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𝑖𝑓 𝑎 + 𝑏 = 𝑐 𝑡ℎ𝑒𝑛 𝑎 + 𝑏 − 𝑏 = 𝑐 − 𝑏, 𝑜𝑟 𝑎 = 𝑐 − 𝑏

As well as it goes for the multiplication property of equality
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𝑖𝑓

𝑎
𝑎
= 𝑐, 𝑎𝑛𝑑 𝑏 ≠ 0, 𝑡ℎ𝑒𝑛
𝑏 = 𝑐 x 𝑏, 𝑜𝑟 𝑎 = 𝑐𝑏
𝑏
𝑏

And naturally this goes for the division property of equality as well
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Anything is acceptable
as long as you do the same thing on both sides
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He now wants to chop it into smaller
pieces
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And then he continues on
into making ten pieces that all are 6ft long before loading them onto his truck
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60=30+30=6+6+6+6+6+6+6+6+6+6
This is called the reflexive property of equality and tells us that any quantity is equal to
itself
𝑎=𝑎
We can also use this example with the pieces of wood to explain the symmetric property
of equality
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𝑖𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑏 = 𝑎
Or if we use our example
𝑖𝑓 60 = 30 + 30, 𝑡ℎ𝑒𝑛 30 + 30 = 60
Another property that can be explained by this is the transitive property of equality
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𝑖𝑓 𝑎 = 𝑏 𝑎𝑛𝑑 𝑏 = 𝑐, 𝑡ℎ𝑒𝑛 𝑎 = 𝑐
Or in the numbers taken from the oak tree example

𝑖𝑓 60 = 30 + 30
𝑎𝑛𝑑 30 + 30 = 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6
𝑡ℎ𝑒𝑛 60 = 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6
Since we know that 30 + 30 = 20 + 40 and that 30 + 30 = 60 we can substitute 30 + 30
for 20 + 40 and get 60 = 20 + 40
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If a = b, then a can be substituted for b in any expression
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Thus we use the common symbols for velocity (v), distance (d)
and time (t) and express it thus:

𝑣=

𝑑
𝑡

We may simply describe a formula as being a variable and an expression separated by
an equal sign between them
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Example
A book club requires a membership fee of $10 in addition to the $2 levied for each
book ordered
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30 = 10 + 2𝑥

C was the cost, i
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it is now $30

30 − 10 = 10 + 2𝑥 − 10

we subtract $10 from each side

20 = 2𝑥

simply

20 2𝑥
=
2
2

divide both sides by 2 to isolate x

10 = 𝑥

x equals to 10

We may purchase 10 books for $30
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You can say that
we put everything else on the other side of the equal sign
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As long as you do the same thing on both sides of the equal
sign you can do whatever you want and in which order you want
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We could have begun by
dividing by 2 instead
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If your equation contains like terms it is preferable to begin by combining the like
terms before continuing solving the equation
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This is done by subtracting 16
from both sides
7𝑥 + 16 − 16 = 30 − 16
7𝑥 = 14

Divide both sides by 7 to isolate the variable
7𝑥 14
=
7
7
𝑥=2

If you have an equation where you have variables on both sides you do basically the
same thing as before
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Before you have worked by first
collecting all constant terms on one side and keep the variable terms on the other side
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You collect all constant terms on one side and the variable
terms on the other side
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e
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Example
𝟒𝒙 + 𝟑 = 𝟐𝒙 + 𝟏𝟏
subtract 2x from both sides
4𝑥 + 3 − 2𝑥 = 2𝑥 + 11 − 2𝑥
Now it looks like any other equation
2𝑥 + 3 = 11
subtract 3 from both sides
2𝑥 + 3 − 3 = 11 − 3
2𝑥 = 8
Divide by 2 on both sides
2𝑥 8
=
2
2
𝑥=4
In the beginning of this section we showed the formula for calculating the velocity where
velocity (v) equals the distance (d) divided by time (t) or
𝑣=

𝑑
𝑡

If we by some chance want to know how far a truck drives in 3 hours at 60 miles per
hour we can use the formula above and rewrite it to solve the distance, d
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This holds true for all formulas and equations
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When we talk about the speed of a car or an
airplane we measure it in miles per hour
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A

ratio is a way to compare two quantities by using division as in miles per hour where
we compare miles and hours
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For instance if one package of cookie mix results in 20 cookies than that would be the
same as to say that two packages will result in 40 cookies
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Example
You know that to make 20 pancakes you have to use 2 eggs
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The cross product is the product of the numerator of one
of the ratios and the denominator of the second ratio
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This is called a scaling
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Scaling involves recreating a model of the
object and sharing its proportions, but where the size differs
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For example, the scale of 1:4 represents a fourth
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If
we wish to calculate the inverse, where we have a 20ft high wall and wish to reproduce it
in the scale of 1:4, we simply calculate:
1
20 x 1: 4 = 20 x = 5
4
In a scale model of 1:X where X is a constant, all measurements become 1/X - of the
real measurement
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Depicting
something in the scale of 2:1 all measurements then become twice as large as in reality
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Similar figures
We often come across two triangles that share the same proportions, i
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the angles are
alike though their sizes differ
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The simplest method of
illustrating two similar triangles is first to draw a triangle, and then to draw a straight line
so that we now has a smaller triangle within the large one
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The ratios of the corresponding sides of
similar figures are equal which means that they are proportions
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Percent means per hundred
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Percent problems are usually solved by using proportions
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How many percent does that
correspond to?
We know that the ratio of girls to all students is
14
21
And we know that this ratio is a proportion to a ratio with the denominator 100
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e
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One of the ratios in these proportions is always a comparison of two numbers (above
14/21)
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The other ratio
is called the rate and always has the denominator 100
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It's calculated as:
𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 =

𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑜𝑟 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒
𝑜𝑙𝑑 𝑎𝑚𝑜𝑢𝑛𝑡

Example
Johnny is at the store where there is a big sign telling him that there is a $4
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99
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99
≈ 0
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99
0
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