Subsidiary Mathematics

My goals

Introduction

An equation is a statement that the values of two mathematical expressions are equal while an inequality is a statement that the values of two mathematical expressions that are not equal. 

Lots of students are interested in aviation. Would you like to be a private pilot and fly your own small plane? You can’t pass the test to get a pilot’s license unless you can work out equations and calculate how to load the plane correctly with people and baggage.  You should already understand what it means to make a profit - you sell an item for more than it cost to make it. So in very simple terms, companies use this equation all the time: Profit = selling price - cost. There are many financial decisions we make everyday based on how much we earn, less all our expenses, then we say we can spend up to x on a new purchase or you put x in a saving account, or something else. These are all inequalities. 

A quadratic equation is an equation of degree 2, meaning that the highest exponent is 2. In daily life, say that you want to throw a ball into the air and have your friend catch it, but you want to give her/him the precise time it will take the ball to arrive. 

To do this, you would use the velocity equation, which calculates the height of the ball based on a parabolic (quadratic) equation. Quadratics are equations and inequalities are useful in calculating areas, figuring out profits, finding speeds,  athletics,… 

Required outcomes 

After completing this unit, the learners should be able to: 

1. Equations and inequalities in one unknown 

Equations

Activity 1

Find the value of x such that the following statements are true; 

To do this, rearrange the given equation such that variables will be in the same side and constants in the other side and then find the value of the variable. 

Example 1 

Solve in set of real numbers: 

Solution

Exercise 1 

Solve in set of real numbers: 

Equations products / quotients

Activity 2

State the method you can use to solve the following equations 

Example 2

Solve in set of real numbers 

Solution

Exercise 2 

Solve in set of real numbers: 

Inequalities

Activity 3

Find the value(s) of x such that the following statements are true 

Recall that

• When the same real number is added or subtracted from each side of the inequality, the direction of the inequality is not changed. 

• The direction of the inequality is not changed if both sides are multiplied or divided by the same positive real number and is reversed if both sides are multiplied or divided by the same negative real number.

Example 3

Solve in set of real numbers: 

Solution

Exercise 3 

Solve the following inequalities 

Inequalities products / quotients

Activity 4 

State the method you can use to solve the following inequalities

Example 4

Solve in set of real numbers; 

Solution 

Exercise 4 

Solve the following inequalities: 

Inequalities involving absolute value 

Activity 5

State the set of all real numbers whose number of units from zero, on number line, are 

1. greater than 4 

2. less than 6 

Hint Draw a number line

Example 5 

Solution

Example 6 

Solution

Example 7 

Solution

Example 8 

A technician measures an electric current which is 0.036 A with a possible error of ±0.002 A. Write this current, i, as an inequality with absolute values.

Solution

The possible error of ±0.002 A means that the difference between the actual current and the value of 0.036 A cannot be more than 0.002 A. 

So the values of i we have can be expressed as: 

Exercise 5 

Solve the following inequalities:

Equations and inequalities in real life problems

Activity 6

How can you do the following?

1. A father is 30 years older than his son. 5 years ago he was four times as old as his son. What is the son’s age? 

2. Betty spent one fifth of her money on food. Then she spent half of what was left for a haircut. She bought a present for 7,000 francs. When she got home, she had 13,000 francs left. How much did Betty have originally?

Equations can be used to solve real life problems. 

To solve real life problems, follow the following steps: 

Example 9

Kalisa is four times as old as his son, and his daughter is 5 years younger than his brother. If their combined ages amount to 73 years, find the age of each person.

Solution 

Example 10

Concrete is a mixture of cement, sand and aggregate. If 4 kg of sand and 6kg of aggregate are used with each kg of cement, how many kg of each are required to make 1,210 kg?

Solution

Example 11

John has 1,260,000 Francs in an account with his bank. If he deposits 30,000 Francs each week into the account, how many weeks will he need to have more than 1,820,000 Francs on his account?

Solution

Example 12

The yield from 50 acres of wheat was more than 1,250 tones. What was the yield per acre?

Solution

Exercise 6

1. The sum of two numbers is 25. One of the numbers exceeds the other by 9. Find the numbers. 

2. The difference between the two numbers is 48. The ratio of the two numbers is 7:3. What are the two numbers? 

3. The length of a rectangle is twice its breadth. If the perimeter is 72 metre, find the length and breadth of the rectangle.

4. Aaron is 5 years younger than Ron. Four years later, Ron will be twice as old as Aaron. Find their present ages. 

5. Sam and Alex play in the same soccer team. Last Saturday Alex scored 3 more goals than Sam, but together they scored less than 9 goals. What are the possible number of goals Alex scored? 

6. Joe enters a race where he has to cycle and run. He cycles a distance of 25 km, and then runs for 20 km. His average running speed is half of his average cycling speed. Joe completes the race in less than 2½ hours, what can we say about his average speeds?

2. Simultaneous equations in two unknowns 

Consider the following system 

Combination (or addition or elimination) method

Activity 7

For each of the following, find two numbers to be multiplied to the equations such that one variable will be eliminated; 

We try to combine the two equations such that we will remain with one equation with one unknown. We find two numbers to be multiplied on each equation and then add up such that one unknown is cancelled.

Exercise 7 

Use elimination method to solve;

Substitution method

Activity 8

In each of the following systems find the value of one variable from one equation and substitute it in the second.

We find the value of one unknown in one equation and put it in another equation to find the value of the remaining unknown. 

Example 17

Solve the following system:

solution

Example 18

solution

Example 19

Solve the following system:

solution

Example 20

Solve the following system: 

This system is a dependent system. Thus, there is an infinity solutions.

Exercise 8 

Use elimination method to solve;

Cramer’s rule (determinants method) 

Activity 9

Find the following determinants. Find the following determinants. 

In order to use Cramer’s rule, x’s must be in the same position and y’s in the same position. 

Consider the following system.

Example 21

Solve the following system: 

solution

Example 22

Solve the following system: 

solution

First rearrange the system such that x’s will be in the same position and y’s will be in the same position. 

Example 23

Solve the following system: 

solution

This system is inconsistent. Thus, there is no solution.

Example 24

solution

Solve the following system: 

This system is a dependent system. Thus, there is an infinity solutions. 

Exercise 9 

Use Cramer’s rule to solve; 

Graphical method 

Activity 10

Given the system 

1. For each equation, choose any two values of x and use them to find values of y; this gives you two points in the form (x,y).

2. Plot the obtained points in xy plane and join these points to obtain the lines. Two points for each equation give one line. 

3. What is the point of intersection for two lines? 

Some systems of linear equations can be solved graphically. To do this, follow the following steps: 

1. Find at least two points for each equation. 

2. Plot the obtained points in xy plane and join these points to obtain the lines. Two points for each equation give one line. 

3. The point of intersection for two lines is the solution for the given system.

Example 25 

Solve the following system by graphical method 

Solution

The two lines intersect at point (3,1). Therefore the solution is  S={(3,1)}.

Example 26 

Solve the following equations graphically if possible

Solution 

We see that the two lines are parallel and do not intersect. Therefore there is no solution. Note that the gradients of the two lines are the same.

Example 27 

Solve the following equations graphically if possible 

Solution

We see that the two lines coincide as a single line. In such case there is infinite number of solutions. 

Exercise 10 

Solve the following system by graphical method 

Solving word problems using simultaneous equations

Activity 11

How can you do the following question? 

Margie is responsible for buying a week’s supply of food and medication for the dogs and cats at a local shelter.  The food and medication for each dog costs twice as much as those supplies for a cat.  She needs to feed 164 cats and 24 dogs.  Her budget is $4240.  How much can Margie spend on each dog for food and medication?

To solve word problems, follow the following steps: 

Example 28

Peter has 23 coins in his pocket. Some of them are 5 Frw coins and the rest are 10 Frw coins. The total value of coins is 205 Frw . Find the number of 10 Frw coins and the number of 5 Frw coins.

Solution

Let x be the number of 10 Frw coins and y be the number of 5 Frw  coins. Then, 

In second equation, 

Thus, there are 18 coins of 10 Frw and 5 coins of 5 Frw .

Example 29

Cinema tickets for 2 adults and 3 children cost 1,200 Frw . The cost for 3 adults and 5 children is 1,900 Frw. Find the cost of an adult ticket and the cost of a child ticket.

Solution

Let x  be the cost of an adult ticket and y be the cost of a child ticket, then 

Thus, the cost of an adult ticket is 300 Frw and the cost of a child ticket is 200 Frw.

Exercise 11

1. A test has twenty questions worth 100 points. The test consists of True/False questions worth 3 points each and multiple choice questions worth 11 points each.  How many multiple choice questions are on the test? 

2. Two small pitchers and one large pitcher can hold 8 cups of water.  One large pitcher minus one small pitcher constitutes 2 cups of water.  How many cups of water can each pitcher hold? 

3. The state fair is a popular field trip destination. This year, the senior class at High School A and the senior class at High School B both planned trips there. The senior class at High School A rented and filled 8 vans and 8 buses with 240 students. High School B rented and filled 4 vans and 1 bus with 54 students. Every van had the same number of students in it as did the buses. Find the number of students in each van and in each bus. 

4. The sum of the digits of a certain two-digit number is 7. Reversing its digits increases the number by 9. What is the number? 

5.  A boat traveled 210 miles downstream and back. The trip downstream took 10 hours. The trip back took 70 hours. What is the speed of the boat in still water? What is the speed of the current?

3. Quadratic equations and inequalities 

Quadratic equations by factorizing or finding square roots

Activity 12

The method of solving quadratic equations by factorization should only be used if is readily factorized by inspection. The method of solving quadratic equations by factorization should only be used if is readily factorized by inspection. 

Example 30

Solution

Example 31 

Solution

Exercise 12 

Solve in set of real numbers the following equations by factorization 

Quadratic equations by completing the square

Activity 13

Before solving quadratic equations by completing the square, let’s look at some examples of expanding a binomial by squaring it.

add a constant to one side of the equation, be sure to add the same constant to the other side of equation.

Example 32 

Solution 

Example 33

Solution 

Exercise 13

Solve in set of real numbers the following equations by completing the square 

Quadratic equations by the formula 

Activity 14

Think about two numbers a and b (a can be equal to b) such that 

Example 34 

Solution 

Example 35 

Solution 

Example 36

Solution 

Example 37 

Solution 

Exercise 14 

Solve in set of real numbers; 

Notice:

Factor form of a quadratic expression

Activity 15

In each of the following, remove brackets and discuss about the result and original form.

Exercise 15 

Find the factor form of 

Equations Reducible to Quadratic Form 

a) Biquadratic equations 

Activity 16

In each of the following rewrite the given equation letting   

Example 41 

Solution 

So, the basic process is to check that the equation is reducible to a quadratic form, then make a quick substitution to turn it into a quadratic equation.  In most cases, to make the check that it’s reducible to quadratic form, all we really need to do is to check that one of the exponents is twice the other.  

Exercise 16 

Solve in set of real numbers 

b) Nested radicals

Activity 17

Example 42   

Solution

Example 43 

Solution

Exercise 17 

Solve in set of real numbers the following equations

c) Irrational equations

Activity 18

Consider the following equation

Irrational equation is the equation involving radical sign. We will see the case the radical sign is a square root. 

To solve an irrational equation, follow these steps: 

Example 44

Solve in set of real numbers

Solution

Example 45 

Solve in set of real numbers 

Solution

Exercise 18 

Solve in set of real numbers the following equations 

Quadratic inequalities

Activity 19

Find the range where 

Example 46 

Solution 

Example 47

Solution

Example 48

Solution 

Example 49

Solution

Example 50

Solution 

Exercise 19 

Solve in set of real numbers 

4. Applications

Activity 20

a)   Supply and demand analysis 

Market equilibrium is when the amount of product produced is equal to the amount of quantity demanded. We can see equilibrium on a graph when the supply function and the demand function intersect, like shown on the graph below. Max can then figure out how to price his new lemonade products based on market equilibrium. 

Example 51 

Assume that in a competitive market the demand schedule is   and the supply schedule is  p=60+0.4q (p=price, q=quantity). If the market is in equilibrium then the equilibrium price and quantity will be where the demand and supply schedules intersect. As this will correspond to a point which is on both the demand schedule and the supply schedule the equilibrium values of  p and q will be such that both equations hold. 

b) Linear motion 

Linear motion is a motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; non uniform linear motion with variable velocity or non-zero acceleration. 

Example 52 

Some examples of linear motion are given below: 

c) Balancing equation 

In chemistry, to balance the chemical equation we set the reactants and products equal to each other. 

Example 53 

d) Calculating Areas 

People frequently need to calculate the area of things like rooms, boxes or plots of land. 

Example 54 

An example might involve building a rectangular box where one side must be twice the length of the other side. So long as this ratio is satisfied, the box will be as a person wants it. Say, however, that a person has only 4 square meters of wood to use for the bottom of the box. Knowing this, the person needs to create an equation for the area of the box using the ratio of the two sides. This means the area (the length times the width) in terms of x would equals  , or  . This equation must be less than or equal to 4 for the person to successfully make a box using his constraints. Once solved, he now knows that one side of the box must be   meters long, and the other must be   meters long.

e) Figuring out a profit 

Sometimes calculating a business’ profit requires using a quadratic function. If you want to sell something (even something as simple as lemonade) you need to decide how many things to produce so that you’ll make a profit. 

Example 55 

Let us say that you’re selling glasses of lemonade, and you want to make 12 glasses. You know, however, that you’ll sell a different number of glasses depending on how you set your price. At 100 francs per glass, you are not likely to sell any, but at 10 francs per glass, you will probably sell 12 glasses in less than a minute. So, to decide where to set your price, use P as a variable. Let’s say you estimate the demand for glasses of lemonade to be at 12 - P. Your revenue, therefore, will be the price times the number of glasses sold: P(12 - P), or  12P–P². Using however much your lemonade costs to produce, you can set this equation equal to that amount and choose a price from there. 

f) Quadratics in Athletics 

In athletic events that involve throwing things, quadratic equations are highly useful. 

Example 56 

Say, for example, you want to throw a ball into the air and have your friend catch it, but you want to give her the precise time it will take the ball to arrive. 

To do this, you would use the velocity equation, which calculates the height of the ball based on a parabolic (quadratic) equation. So, say you begin by throwing the ball at 3 meters, where your hands are. Also assume that you can throw the ball upward at 14 meters per second, and that the earth’s gravity is reducing the ball’s speed at a rate of 5 meters per second squared. This means that we can calculate the height, using the variable t for time, in the form of  h=3+14t –t² . If your friend’s hands are also at 3 metres in height, how many seconds will it take the ball to reach her? To answer this, set the equation equal to 3 = h, and solve for t. The answer is approximately 2.8 seconds.

g) Finding a Speed 

Quadratic equations are also useful in calculating speeds. Avid kayakers, for example, use quadratic equations to estimate their speed when going up and down a river. 

Example 57 

Assume a kayaker is going up a river, and the river moves at 2 km/hr. Say he goes upstream -- against the current -- at 15 km, and the trip takes him 3 hours to go there and return. Remember that time = distance / speed. Let v = the kayak’s speed relative to land, and let x = the kayak’s speed in the water. So, we know that, while traveling upstream, the kayak’s speed is v = x - 2 (subtract 2 for the resistance from the river current), and while going downstream, the kayak’s speed is v = x + 2. 

Unit summary 

Balancing equation In chemistry, to balance the chemical equation we set the reactants and products equal to each other. 

Calculating Areas People frequently need to calculate the area of things like rooms, boxes or plots of land. 

Figuring Out a Profit Sometimes calculating a business’ profit requires using a quadratic function. If you want to sell something (even something as simple as lemonade) you need to decide how many things to produce so that you’ll make a profit. 

Quadratics in Athletics In athletic events that involve throwing things, quadratic equations are highly useful. 

Finding a Speed Quadratic equations are also useful in calculating speeds. Avid kayakers, for example, use quadratic equations to estimate their speed when going up and down a river.