# 3.5 Mathematical models

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## Introduction

Tom and Jane are friends. Tom picked up Jane's Physics test paper, but will not tell Jane what her marks are. He knows that Jane hates maths so he decided to tease her. Tom says: “I have 2 marks more than you do and the sum of both our marks is equal to 14. How much did we get?”

Let's help Jane find out what her marks are. We have two unknowns, Tom's mark (which we shall call $t$ ) and Jane's mark (which we shall call $j$ ). Tom has 2 more marks than Jane. Therefore,

$t=j+2$

Also, both marks add up to 14. Therefore,

$t+j=14$

The two equations make up a set of linear (because the highest power is one) simultaneous equations, which we know how to solve! Substitute for $t$ in the second equation to get:

$\begin{array}{ccc}\hfill t+j& =& 14\hfill \\ \hfill j+2+j& =& 14\hfill \\ \hfill 2j+2& =& 14\hfill \\ \hfill 2\left(j+1\right)& =& 14\hfill \\ \hfill j+1& =& 7\hfill \\ \hfill j& =& 7-1\hfill \\ & =& 6\hfill \end{array}$

Then,

$\begin{array}{ccc}\hfill t& =& j+2\hfill \\ & =& 6+2\hfill \\ & =& 8\hfill \end{array}$

So, we see that Tom scored 8 on his test and Jane scored 6.

This problem is an example of a simple mathematical model . We took a problem and we were able to write a set of equations that represented the problem mathematically. The solution of the equations then gave the solution to the problem.

## Problem solving strategy

The purpose of this section is to teach you the skills that you need to be able to take a problem and formulate it mathematically in order to solve it. The general steps to follow are:

1. Read ALL of it !
2. Find out what is requested.
3. Use a variable(s) to denote the unknown quantity/quantities that has/have been requested e.g., $x$ .
4. Rewrite the information given in terms of the variable(s). That is, translate the words into algebraic expressions.
5. Set up an equation or set of equations (i.e. a mathematical sentence or model) to solve the required variable.
6. Solve the equation algebraically to find the result.

## Application of mathematical modelling

Three rulers and two pens have a total cost of R 21,00. One ruler and one pen have a total cost of R 8,00. How much does a ruler costs on its own and how much does a pen cost on its own?

1. Let the cost of one ruler be $x$ rand and the cost of one pen be $y$ rand.

2. $\begin{array}{ccc}\hfill 3x+2y& =& 21\hfill \\ \hfill x+y& =& 8\hfill \end{array}$
3. First solve the second equation for $y$ :

$y=8-x$

and substitute the result into the first equation:

$\begin{array}{ccc}\hfill 3x+2\left(8-x\right)& =& 21\hfill \\ \hfill 3x+16-2x& =& 21\hfill \\ \hfill x& =& 5\hfill \end{array}$

therefore

$\begin{array}{ccc}\hfill y& =& 8-5\hfill \\ \hfill y& =& 3\hfill \end{array}$
4.             One ruler costs R 5,00 and one pen costs R 3,00.

A fruit shake costs R2,00 more than a chocolate milkshake. If three fruit shakes and 5 chocolate milkshakes cost R78,00, determine the individual prices.

1. Let the price of a chocolate milkshake be $x$ and the price of a fruitshake be $y$ .

 Price number Total Fruit $y$ 3 $3y$ Chocolate $x$ 5 $5x$
2. $\begin{array}{ccc}\hfill 3y+5x& =& 78\hfill \end{array}$

$y=x+2$

3. $\begin{array}{ccc}\hfill 3\left(x+2\right)+5x& =& 78\hfill \\ \hfill 3x+6+5x& =& 78\hfill \\ \hfill 8x& =& 72\hfill \\ \hfill x& =& 9\hfill \\ \hfill y& =& x+2\hfill \\ \hfill & =& 9+2\hfill \\ \hfill & =& 11\hfill \end{array}$
4. One chocolate milkshake costs R 9,00 and one Fruitshake costs R 11,00

## Mathematical models

1. Stephen has 1 l of a mixture containing 69% of salt. How much water must Stephen add to make the mixture 50% salt? Write your answer as a fraction of a litre.
2. The diagonal of a rectangle is 25 cm more than its width. The length of the rectangle is 17 cm more than its width. What are the dimensions of the rectangle?
3. The sum of 27 and 12 is 73 more than an unknown number. Find the unknown number.
4. The two smaller angles in a right-angled triangle are in the ratio of 1:2. What are the sizes of the two angles?
5. George owns a bakery that specialises in wedding cakes. For each wedding cake, it costs George R150 for ingredients, R50 for overhead, and R5 for advertising. George's wedding cakes cost R400 each. As a percentage of George's costs, how much profit does he make for each cake sold?
6. If 4 times a number is increased by 7, the result is 15 less than the square of the number. Find the numbers that satisfy this statement, by formulating an equation and then solving it.
7. The length of a rectangle is 2 cm more than the width of the rectangle. The perimeter of the rectangle is 20 cm. Find the length and the width of the rectangle.

## Summary

• Linear equations A linear equation is an equation where the power of the variable (letter, e.g. x) is 1(one).Has at most one solution
• Quadratic equations A quadratic equation is an equation where the power of the variable is at most 2.Has at most two solutions
• Exponential equations Exponential equations generally have the unknown variable as the power.ka^(x+p) = m Equality for Exponential FunctionsIf a is a positive number such that a>0, then: a^x = a^yif and only if: x=y
• Linear inequalities A linear inequality is similar to a linear equation and has the power of the variable equal to 1.When you divide or multiply both sides of an inequality by any number with a minus sign, the direction of the inequality changes. Solve as for linear equations
• Linear simultaneous equations When two unknown variables need to be solved for, two equations are required and these equations are known as simultaneous equations.Graphical or algebraic solutions Graphical solution: Draw the graph of each equation and the solution is the co-ordinates of intersectionAlgebraic solution: Solve equation one, for variable one and then substitute it into equation two.
• Mathematical models Take a problem, write equations that represent it, solve the equations and that solves the problem.

## End of chapter exercises

1. What are the roots of the quadratic equation ${x}^{2}-3x+2=0$ ?
2. What are the solutions to the equation ${x}^{2}+x=6$ ?
3. In the equation $y=2{x}^{2}-5x-18$ , which is a value of $x$ when $y=0$ ?
4. Manuel has 5 more CDs than Pedro has. Bob has twice as many CDs as Manuel has. Altogether the boys have 63 CDs. Find how many CDs each person has.
5. Seven-eighths of a certain number is 5 more than one-third of the number. Find the number.
6. A man runs to a telephone and back in 15 minutes. His speed on the way to the telephone is 5 m/s and his speed on the way back is 4 m/s. Find the distance to the telephone.
7. Solve the inequality and then answer the questions: $\frac{x}{3}-14>14-\frac{x}{4}$
1. If $x\in \mathbb{R}$ , write the solution in interval notation.
2. if $x\in \mathbb{Z}$ and $x<51$ , write the solution as a set of integers.
8. Solve for $a$ : $\frac{1-a}{2}-\frac{2-a}{3}>1$
9. Solve for $x$ : $x-1=\frac{42}{x}$
10. Solve for $x$ and $y$ : $7x+3y=13$ and $2x-3y=-4$

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years? Kala Reply lim x to infinity e^1-e^-1/log(1+x) given eccentricity and a point find the equiation Moses Reply 12, 17, 22.... 25th term Alexandra Reply 12, 17, 22.... 25th term Akash College algebra is really hard? Shirleen Reply Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table. Carole I'm 13 and I understand it great AJ I am 1 year old but I can do it! 1+1=2 proof very hard for me though. Atone hi Adu Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily. Vedant hi vedant can u help me with some assignments Solomon find the 15th term of the geometric sequince whose first is 18 and last term of 387 Jerwin Reply I know this work salma The given of f(x=x-2. then what is the value of this f(3) 5f(x+1) virgelyn Reply hmm well what is the answer Abhi If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10 Augustine how do they get the third part x = (32)5/4 kinnecy Reply make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be AJ how Sheref can someone help me with some logarithmic and exponential equations. Jeffrey Reply sure. what is your question? ninjadapaul 20/(×-6^2) Salomon okay, so you have 6 raised to the power of 2. what is that part of your answer ninjadapaul I don't understand what the A with approx sign and the boxed x mean ninjadapaul it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared Salomon I'm not sure why it wrote it the other way Salomon I got X =-6 Salomon ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6 ninjadapaul oops. ignore that. ninjadapaul so you not have an equal sign anywhere in the original equation? ninjadapaul hmm Abhi is it a question of log Abhi 🤔. Abhi I rally confuse this number And equations too I need exactly help salma But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends salma Commplementary angles Idrissa Reply hello Sherica im all ears I need to learn Sherica right! what he said ⤴⤴⤴ Tamia hii Uday hi salma hi Ayuba Hello opoku hi Ali greetings from Iran Ali salut. from Algeria Bach hi Nharnhar A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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