Mathematics summary
Chapter one: Linear Relationships
- Linear equations
Solving an equation by ensuring that the variables only appear on the left-hand side:
10x-4=7x+20
10x-7x=20+4
When you move terms to the other side of the = sign, negative numbers become positive and positive numbers become negative.
How to solve linear equations:
- Multiply out the brackets
- All terms containing x to the left-hand side and the rest to the right-hand side
- Simplify both sides
- Divide by the number in front of the x.
- Inequalities
4(a-3) ≥ 4-3(5-a) This is a linear inequality.
4a-12 ≥ 4-15+3a
4a-3a≥4-15+12
a ≥ 1
When you divide by a negative number, the > and < symbols are flipped.
Solving an inequality works the same as solving a linear equation. Except that the last step could be to flip the < and > symbol.
X2 > 16 is a quadratic inequality. It results in x < -4 or x > 4.
The solutions to x2 < 16 lie between -4 and 4.
X lies between -4 and 4.
-4 < x < 4.
Leave square roots such as √2 as they are.
X2 < -16 no solutions X2 > -16 any x has a solution.
X2 ≤ -16 no solutions X2 ≥ -16 any x is a solution.
- Linear formulas
If there is a linear relationship between x and y, it will be in the form of y=ax+b.
- The graph is a straight line.
- If you go 1 step to the right, you will go up a steps.
- The point of intersection with the y-axis is (0,b), so the y intercept is b.
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 l: y = -0.25x + 2. Point of intersection is A(0,2) on the y axis. Then use;
- X = 4 results in y = -25 x 4 + 2 = 1. Therefore B(4,1).
- Or a = -0.25 means 1 to the right and 0.25 down. For example, 4 to the right and 1 down.
How to generate a formula for a line:
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.
A = Vertical : Horizontal.
Lines l : y = 2x + 3 and m : y = 2x -8 are parallel because a is the same in both formulas.
For example:
Point A(4, -5) lies on line m : y = -3x + b. Calculate b.
How to work it out:
M : y = -3x + b
A (4, -5) on m. à -3 x 4 + b = -5.
-12 + b = -5.
b = -5 + 12.
b = 7.
Generate the formula for line l which is parallel to line m : y = 5x – 1 and passes through point B(3,8).
How to work it out:
You know that l : y = ax+b.
l is parallel to m : y = 5x – 1, therefore a = 5.
The result is l : y = 5x + b
B(3,8) on l. à 5 x 3 + b = 8.
15 + b = 8.
b = 8-15
b = -7.
Therefor l : y = 5x – 7.
- Linear Functions
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 f is given by x à 5x – 12. The function value of 3 is f(3) = 5 x 3 – 12 = 15 – 12 = 3. The function value of a random x is f(x) = 5x – 12. We call f(x) = 5x – 12 the brackets notation of f.
Brackets notation: f(x) = 3x + 1.
Y = 3x + 1.
Functions such as f(x) = 3x – 1, g(x) = -x + 5 and h(x) = 5x are examples of linear functions. General form of a linear equation: f(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).
- Sum and difference graphs
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.
