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Mathematics summary
Chapter one: Linear Relationships
<ol>
<li>Linear equations</li>
</ol>
<p style="margin-left:18.0pt">Solving an equation by ensuring that the variables only
appear on the left-hand side:
<p style="margin-left:18.0pt">10x-4=7x+20
<p style="margin-left:18.0pt">10x-7x=20+4
<p style="margin-left:18.0pt">When you move terms to the other side of the = sign,
negative numbers become positive and positive numbers become negative.
<p style="margin-left:18.0pt">How to solve linear equations:
<ol>
<li>Multiply out the brackets</li>
<li>All terms containing x to the left-hand side and the rest to the
right-hand side</li>
<li>Simplify both sides</li>
<li>Divide by the number in front of the x.</li>
</ol>
<ol>
<li>Inequalities</li>
</ol>
<p style="margin-left:18.0pt">4(a-3) ≥ 4-3(5-a)
linear inequality.
This is a
<p style="margin-left:18.0pt">4a-12 ≥ 4-15+3a
<p style="margin-left:18.0pt">4a-3a≥4-15+12
<p style="margin-left:18.0pt">a ≥ 1
<p style="margin-left:18.0pt">When you divide by a negative number, the > and <
symbols are flipped.
<p style="margin-left:18.0pt">Solving an inequality works the same as solving a
linear equation. Except that the last step could be to flip the < and > symbol.
<p style="margin-left:18.0pt">X
<sup style="line-height:1.5em">2 > 16 is a quadratic inequality. It results in x < -4
or x > 4.
<p style="margin-left:18.0pt">The solutions to x2 < 16 lie between -4 and 4.
<p style="margin-left:18.0pt">X lies between -4 and 4.
<p style="margin-left:18.0pt">-4 < x < 4.
<p style="margin-left:18.0pt">Leave square roots such as √2 as they are.
<p style="margin-left:18.0pt">X
<sup style="line-height:1.5em">2 < -16 no solutions
<sup style="line-height:1.5em">2 > -16 any x has a solution.
<p style="margin-left:18.0pt">X2 ≤ -16 no solutions
-16 any x is a solution.
X
X2 ≥
<ol>
<li>Linear formulas</li>
</ol>
<p style="margin-left:18.0pt">If there is a linear relationship between x and y, it will be
in the form of y=ax+b.
<ul>
<li>The graph is a straight line.</li>
<li>If you go 1 step to the right, you will go up a steps.</li>
<li>The point of intersection with the y-axis is (0,b), so the y intercept is
b.</li>
</ul>
When N=0.75t+1. The t-axis is the horizontal axis and the N-axis the vertical one.
The graph intersects the N-axis (0,1). If you go 1 step to the right, you must go up
0.75 steps.
Draw line
<em style="line-height:1.5em">l</em>: y = -0.25x + 2. Point of intersection is A(0,2)
on the y axis. Then use;
<ul>
<li> X = 4
results in y = -25 x 4 + 2 = 1. Therefore B(4,1).</li>
<li>Or a = -0.25 means 1 to the right and 0.25 down. For example, 4 to
the right and 1 down.</li>
</ul>
<p style="margin-left:36.0pt">How to generate a formula for a line:
<p style="margin-left:36.0pt">You start with y=ax+b. b is the point of intersection with
the y axis. Then select two coordinates of a grid point and divide them.
<p style="margin-left:36.0pt">A = Vertical : Horizontal.
<p style="margin-left:36.0pt">Lines
<em style="line-height:1.5em">l</em> : y = 2x + 3 and
<em style="line-height:1.5em">m</em> : y = 2x -8 are parallel because a is the same
in both formulas.
<p style="margin-left:36.0pt">For example:
<p style="margin-left:36.0pt">Point A(4, -5) lies on line m : y = -3x + b. Calculate b.
<p style="margin-left:36.0pt">How to work it out:
<p style="margin-left:36.0pt">M : y = -3x + b
<p style="margin-left:36.0pt">A (4, -5) on m. à -3 x 4 + b = -5.
<p style="margin-left:36.0pt">
-12 + b = -5.
<p style="margin-left:36.0pt">
b = -5 + 12.
<p style="margin-left:36.0pt">
b = 7.
<p style="margin-left:36.0pt">Generate the formula for line
<em style="line-height:1.5em">l</em> which is parallel to line
<em style="line-height:1.5em">m</em> : y = 5x – 1 and passes through point B(3,8).
<p style="margin-left:36.0pt"><em style="line-height:1.5em">How to work it
out:</em>
<p style="margin-left:36.0pt">You know that l : y = ax+b.
<p style="margin-left:36.0pt">l is parallel to m : y = 5x – 1, therefore a = 5.
<p style="margin-left:36.0pt">The result is l : y = 5x + b
<p style="margin-left:36.0pt">B(3,8) on l. à 5 x 3 + b = 8.
<p style="margin-left:36.0pt">
15 + b = 8.
<p style="margin-left:36.0pt">
b = 8-15
<p style="margin-left:36.0pt">
b = -7.
<p style="margin-left:36.0pt">Therefor l : y = 5x – 7.
<ol>
<li>Linear Functions</li>
</ol>
In 12 à 32, 12 is called the argument and 32 is the image. The arrow points from the
argument to the image. Such a machine is called a function.
2x + 8 : x à 2x + 8.
Another one: x à -2x + 6. For this function, the image 5 is equal to -2 x 5 + 6 = -10
+ 6 = -4. Therefore 5 à -4.
With functions, we call the argument x and the image y.
So the function x à 2x + 5 means the same as the formula y = 2x + 5.
Let’s name the function f. The image of 4 is equal to 2 x 4 + 5 = 13. f(4) = 13.
Function
<em style="line-height:1.5em">f </em>is given by x à 5x – 12. The function value of
3 is
<em style="line-height:1.5em">f</em>(3) = 5 x 3 – 12 = 15 – 12 = 3. The function
value of a random x is
<em style="line-height:1.5em">f</em>(x) = 5x – 12. We call
<em style="line-height:1.5em">f</em>(x) = 5x – 12 the brackets notation of
<em style="line-height:1.5em">f.</em>
Brackets notation:
<em style="line-height:1.5em">f</em>(x) = 3x + 1.
Y = 3x + 1.
Functions such as
<em style="line-height:1.5em">f</em>(x) = 3x – 1,
<em style="line-height:1.5em">g</em>(x) = -x + 5 and
<em style="line-height:1.5em">h</em>(x) = 5x are examples of linear functions.
General form of a linear equation:
<em style="line-height:1.5em">f</em>(x) = ax+b.
For the graph of function f the following applies:
x-intercept
The y-coordinate is 0.
The x-coordinate follows from f(x) = 0.
The x-intercept is the solution to f(x) = 0.
y-intercept
The x-coordinate is 0.
The y-coordinate is f(0)
Therefore the y-intercept is f(0).
The x-coordinate follows from f(x) = g(x).
The y-coordinate is found by filling in the solution on f(x) or g(x).
<ol>
<li>Sum and difference graphs</li>
</ol>
When you add up 2 graphs, the new graph is called the sum graph. Then you can also
draw the difference graph.
You only need two points to draw a sum graph when the sum graph is a straight
line.
The sum graph of two lines is a straight line. When drawing it, you can use the
points where each of the graphs intersect the x-axis.
If you know the formulas of two graphs, you can easily work out the formula of the
sum graph. If the formula for graph I is y = 0.5x + 1, and the formula for graph II
is y = -x + 2, then the formula for the sum graph is y = 0.5x + 1 + -x + 2, or y = -0.5x
+ 3.
There are 2 possibilities for the difference graph of graphs I: y = 0.5x + 1 and II : y
= -x + 2. You can consider the difference graph I – II, but also the difference graph
II – I:
Y = 0.5x + 1 – (-x + 2), therefore y = 0.5x + 1 + x – 2, or y = 1.5x – 1.
For difference graph II – I:
Y = -x + 2 – (0.5x + 1), therefore y = -x + 2 – 0.5x – 1, or y = -1.5x – 1.