RS-AV: Matematik Lektioner

 

Exponents Exponents are used to express very large numbers in a simplified form. ___ seconds = 1 minute 60 ___ minutes = 1 hour 60 ___ seconds = 1 hour 60*60=3,600 ___ hours = 1 day 24 ___ minutes = 1 day 24*60=1,440 ___ seconds = 1 day 24*60*60=86,400 ___ days = 1 week 7 ___ seconds = 1 week 7*86,400=604,800 ___ days = 1 month 30 ___ seconds = 1 month 30*86,400=2,592,000 ___ days = 1 year 365 ___ seconds = 1 year 365*86,400=31,536,000 Speed: On foot: 6 kilometers in 1 hour Running: 12 km/hour Bicycle: 20 km/hour Scooter: 60 km/hour Train: 100 km/hour Car: 140 km/hour High-end car: 250 km/hour Sports car: 350 km/hour Airplane: 700 km/hour Supersonic airplane: 1200 km/hour LIGHT 300,000 km in 1 second 300,000*60 km in 1 minute 300,000*60*60 km in 1 hour 300,000*60*60*24 km in 1 day 300,000*60*60*24*365 km in 1 year 94,608,000,000,000 Ninety-four thousand six hundred and eight billion kilometers in a ...

(x/3) + (4/5) = 1/4 x/3 = (1/4) - (4/5) [While performing addition or subtraction on fractions, the key is to make the denominators the same.] x/3 = (5/20) - (16/20) x/3 = -11/20 x = (-11/20) * 3 x= -33/20 F+F+F=15 3F=15 F=15/3=5 R-4(F)=-5 R-4(5)=-5 R-20=-5 R=-5+20 [5 steps left, 20 steps right] R=15 Bob's Age = x Matt's Age = x+4 Lana's Age = 2(x+4)-5 =2x+2*4-5 =2x+8-5 =2x+3 Multiplication is distributive over addition. a(b+c) = ab+ac This is known as DISTRIBUTIVE PROPERTY. (2/3)(5+3x-11) =(2/3)*5 + (2/3)*3x - (2/3)*11 =(10/3) + 2x - (22/3) =2x-(12/3) =2x-4 Let's say length is l and breadth is b. Perimeter = 2l + 2b Area = lb Perimeter is equal to thrice the area. 2l+2b = 3lb b=1 cm 2l+2*1 = 3l*1 2l+2=3l 3l=2l+2 3l-2l=2 l=2 cm -9-8 9 steps left, again 8 steps left -17 No. of questions faced=20 Let's say, x answers were correct and y answers were wrong. x+y=20..............(1) For x correct answers, he gets 5x points, and, for y wrong answers, he gets -3y points. 5x...

How far is the point (5,10) from Origin? Suppose x is the distance from Origin to (5,10). Here, we can imagine a right triangle with the vertices at (0,0), (5,0) and (5,10). That is, the floor distance is 5 and the wall height is 10. 5 and 10 are the arms of the right triangle here. We want to find the hypotenuse. According to Pythagoras Theorem, we can write x^2=5^2+10^2 x^2=25+100 x^2=125 x= √125 √

What is the slope of (x/2) + (y/3) = 1? 3/2 Does this line make an acute angle with the X-Axis or obtuse? Obtuse Does the point (1,3/2) lie on this line? (x/2)+(y/3)=1 (1/2)+(3/2 divided by 3)=1 (1/2)+(1/2)=1 1=1 LHS=RHS Yes, the point lies on the line.

SOLVING LINEAR EQUATIONS IN TWO VARIABLES Example 1: x+y=5 x-y=1 Where do these two lines intersect? Let's add the two equations. (x+y)+(x-y)=5+1 x+y+x-y=6 2x=6 x=6/2=3 x+y=5 3+y=5 y=5-3=2 (3,2) is the point of intersection of the given lines. Example 2: 2x+3y=14 5x-y=1 Where do the two lines cut? 2x+3y+5x-y=14+1 7x+2y=15 This isn't working. Again, let's try differently. 2x+3y=14 5x-y=1 Let's multiply both sides of this equation by 3. 15x-3y=3 Adding the two equations, 2x+3y+15x-3y=14+3 17x=17 x=17/17=1 5x-y=1 5(1)-y=1 5-y=1 -y=1-5 -y=-4 y=4 (1,4) is the point of intersection of the given lines. Example 3: 2x+5y+29=0 3x+2y+16=0 Let's multiply the first equation by 2 and the second equation by 5. 4x+10y+58=0 15x+10y+80=0 Subtracting the second equation from the first one, (4x+10y+58)-(15x+10y+80)=0 4x+10y+58-15x-10y-80=0 4x-15x+58-80=0 -11x-22=0 -11x=22 x==22/-11=-22/11=-2 2x+5y+29=0 2(-2)+5y+29=0 -4+5y+29=0 5y+25=0 5y=-25 y=-25/5=-5 (-2,-5) is the point of intersect...

Ax+By+C=0 Does this represent a straight line? Yes, it does because it is a "LINEAR" equation in x and y. (The power of x is 1 and the power of y is 1.) What is the slope of this line? What is the y-intercept of this line? And what is the x-intercept? By = -Ax-C y = (-Ax-C)/B y = (-Ax/B) - (C/B) y = (-A/B)x - (C/B) Compare this to y = mx+c m = -A/B c = C/B Therefore, Ax+By+C=0 has a slope of -A/B. Its y-intercept is -C/B. It cuts the Y-Axis at the point (0, -C/B). To find the x-intercept, we have to substitute 0 for y. Ax+By+C=0 Ax+B(0)+C=0 Ax+C=0 Ax=-C x = -C/A So the x-intercept is -C/A. That is, Ax+By+C=0 cuts the X-Axis at the point (-C/A,0). Summary: Ax+By+C=0, Slope=-A/B, y-intercept=-C/B, x-intercept=-C/A What is the slope of any line parallel to Ax+By+C=0? -A/B What is the slope of any line perpendicular to Ax+By+C=0? Let us say the slope of the perpendicular line is p. (-A/B)p=-1 p=(-1)(-B/A) p=B/A Analyse 2x+3y+4=0. [Comparing with Ax+By+C=0, A=2, B=3 and C=4.] 01) ...

A line makes x-intercept 2 and y-intercept 3. What is its equation? A line passes through (2,5) and (4,9). What is its equation? A line passes through (1,1) and (2,2). What is its equation? Four ways of writing the equation: 1. Slope-intercept form 2. Point form 3. Intercept form 4. Observing the pattern of coordinates (1,1), (2,2) y=x (1,-1), (2,-2) y=-x (1,2), (2,3), (3,4) y=x+1 (1,0), (2,1), (3,2) y=x-1 (1,3), (2, 5), (4,9) y=2x+1 (2,3), (3,5), (4,7) y=2x-1 (2,4), (3,6), (4,8) y=2x (3,1), (6,2), (9,3) y=x/3 or 3y = x If slope is positive, it's a line leaning against the right wall. If slope is negative, it's a line leaning against the left wall. A constant term in the equation indicates the line cuts the y-axis somewhere. If there is no constant in the equation, it means the line passes through the origin. If the slope of two line is the same, they are parallel. If the product of the slopes of two lines equals -1, they are perpendicular. (1)y=2x (2)y=2x+3 (1) passes through ...

Say (x1,y1) and (x2,y2) are two points on a straight line. Consider some general point (x,y) on the line. Let's measure slope using (x1,y1) and (x,y). Slope = (y-y1)/(x-x1) Again, let's meaure slope using the two given points (x1,y1) and (x2,y2). Now, slope = (y2-y1)/(x2-x1) But we know  (y-y1)/(x-x1) must be equal to  (y2-y1)/(x2-x1). Therefore,  (y-y1)/(x-x1) =  (y2-y1)/(x2-x1) This is the equation of the line. This is called POINT FORM. Suppose, a line cuts the X-Axis at (a,0) and the Y-Axis at (0,b). Say, (x,y) is some general point on the line. Now, (y-0)/(x-a)=(b-0)/(0-a) y/(x-a)=-b/a If this is simplified, we get (x/a)+(y/b)=1 This is called INTERCEPT FORM of a straight line.

Given the equation of a line, to find its slope, compare the given equation to y=mx+c. Whatever is the coefficient of x, it is the slope of the line. In the given equation, if the LHS is not y, you have to modify the given equation and bring it into the form y=mx+c. Given the equation of a line, to find its y-intercept, compare the given equation to y=mx+c. Whatever is the constant term, it is the y-intercept made by the line. Alternatively, we can substitute 0 for x, because any point on the Y-Axis will have its x-coordinate as zero. Given the equation of a line, to find its x-intercept, we can substitute 0 for y, because any point on the X-Axis will have its y-coordinate as zero. y=mx+c is called the slope-intercept form of a straight line. +a in the LHS becomes –a in the RHS. –a in the LHS becomes +a in the RHS. *a in the LHS becomes /a in the RHS. /a in the LHS becomes *a in the RHS.

When a line passes through the origin, at any point on the line, the y-coordinate is equal to the wall height and the x-coordinate is equal to the floor distance. If the line cuts the positive Y-Axis, at any point on the line, the wall height is less than the y-coordinate of the point. If the line cuts the negative Y-Axis, at any point on the line, the wall height is greater than the y-coordinate of the point. In these two cases, the x-coordinate is still equal to the floor distance. y=mx+c represents a straight line whose slope is m and y-intercept is c. If the line passes through the origin, c=0, so y=mx.

A line passes through the origin. (3,10) is a point on the line. What is the equation of this line? (0,0) and (3,10) are two points on the line. Slope = difference between y-coordinates/difference between x-coordinates = (10-0)/(3-0) = 10/3 Slope is the same thing as y/x here. y/x = 10/3 By eliminating x in LHS, we get y=10x/3 This is the equation of the given line. Therefore, if a line passes through the origin, y equals slope times x. Slope is denoted by m. y=mx

We can find the equation of a line by observing the relationship pattern between the x and y coordinates. The equation of the line connecting (0,0), (1,1) and (2,2) is y=x. (In math convention, y is always the LHS of the equation of a line. Also, y and x are written in the lower case.) The equation of the line connecting (1,-1), (2,-2) (3,-3) is y=-x. The equation of the line connecting (1,2), (2,4) (3,6) is y=-2x. The equation of the line connecting (2,1), (4,-2) (6,3) is y=x/2. The equation of the line connecting (1,3), (2,5) (3,7) is y=2x+1. The equation of the line connecting (1,1), (2,3) (3,5) is y=2x-1. The slope of a straight line is the difference between the y-coordinates divided by the difference between the x-coordinates, that is, the height of the ladder above the floor divided by the distance of the foot from the wall. If the line is imagined as a ladder, the foot of which is at (a,b) and the top is at (c,d), the slope of the ladder is (d-b)/(c-a).

Slanting lines have both x and y in their equations. A slanting line may be imagined as a ladder resting against a wall. Suppose (a,b) is and (c,d) are the ends of the ladder. Say (a,b) is the foot of the ladder. It's on the floor. Say (c,d) is the top of the ladder. It's in contact with the wall. Now (a,b) is the lower point and (c,d) is the higher point. y=b is the equation of the floor. It's on the vertical x=a. x=c is the equation of the wall. It's on the horizontal y=d. The straight distance between (a,b) and (c,d) is the length of the ladder.

If you want to count a number of vertical bars pointing each of them with your index finger, your hand movement will be horizontal. That is, the X-coordinate is a count of the verticals. If you want to count a number of horizontal bars pointing each of them with your index finger, your hand movement will be vertical. That is, the Y-coordinate is a count of the horizontals. All along the X-Axis, the Y-coordinate is zero. Therefore, the equation of the X-Axis is y=0. All along the Y-Axis, the X-coordinate is zero, Therefore, the equation of the Y-Axis is x=0. We cannot write an equation for the Origin because it is a point and not a line. Equations exist only for lines and not for points. A horizontal line 1 unit above the X-Axis has the equation y=1 and a horizontal line 1 unit below the X-Axis has the equation y=-1. Any horizontal line in the Cartesian Plane has the equation y=k. A vertical line 1 unit to the right of the Y-Axis has the equation x=1 and a vertical line 1 unit to the le...

If the world is a Cartesian plane, North America is in II Quadrant. South America is in III Quadrant. Africa is in I and II Quadrants. Australia is in IV Quadrant. Europe is in I Quadrant. Asia is in I Quadrant. Antarctica is in III and IV Quadrants. Longitude is the X-coordinate of a place and its latitude is the Y-coordinate. A point (m,n) is on the m th vertical and n th horizontal. To move from (2,5) to (5,2) we have to move three units east and three units north.

Equator is the imaginary horizontal line around the middle of Earth. Prime Meridian, also called the Greenwich Longitude, is the imaginary vertical line around the middle of Earth. When it's 5:30 a.m. in India, it's 2 a.m. in Sweden and it's still midnight in London. Equator and Prime Meridian are like the X-Axis and Y-Axis in math. X and Y Axes are at right angles to each other and their point of intersection is called the Origin. Up, down, right and left are similar to north, south, east and west respectively. X and Y Axes divide a plane into 4 parts called quadrants ordered anti-clockwise. The four quadrants may be likened to northeast, northwest, southwest and southeast respectively. The quadrants make up the Cartesian Plane. The Origin is a point which has 0 dimension. The two axes are lines which have 1 dimension called 'length'. The plane has two dimensions namely length and breadth. The part of X-Axis to the right of the Origin is called the positive X-Axis ...