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Mathematics summary Chapter one: Linear Relationships 1.1 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: 1. Multiply out the brackets 2. All terms containing x to the left-hand side and the rest to the right-hand side 3. Simplify both sides 4. Divide by the number in front of the x. 1.2 Inequalities 4(a-3) ≥ 4-3(5-a) 4a-12 ≥ 4-15+3a 4a-3a≥4-15+12 a≥1 This is a linear inequality. 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 no solutions X2 > -16 any x has a solution. X2 ≥ -16 any x is a solution. 1.3 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. -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. 5 x 3 + b = 8. 15 + b = 8. b = 8-15 b = -7. Therefor l : y = 5x – 7. 1.4 Linear Functions In 12 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 2x + 8. Another one: x -2x + 6. For this function, the image 5 is equal to -2 x 5 + 6 = -10 + 6 = -4. Therefore 5 -4. With functions, we call the argument x and the image y. So the function x 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 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 y-intercept The y-coordinate is 0. The x-coordinate follows from f(x) = 0. The x-intercept is the solution to f(x) = 0. 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). 1.5 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.