Maths Assignment - Linear Equations and Applications, University

This comprehensive maths assignment solution covers various aspects of linear equations. The solution starts with finding intercepts and graphing linear equations. It then delves into word problems involving cost analysis, revenue, and break-even points, including scenarios about car rentals, sales...Read More

Running head: MATHS
MATHS
Name of the Student
Name of the University
Author Note

2
MATHS
Activity 10:
Part A:
1. 3x+ y = 1
x intercept = (1/3,0)
y intercept = (0,1)
2.
2x+5y = 10
X intercept = (5,0)
Y intercept = (0,2)

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MATHS
3. 4x -5y + 15 = 0
X intercept = (-15/4,0)
Y intercept = (0,3)
4. x+2y = 6
X intercept = (6,0)
Y intercept = (0,3)
5. 4x+3y = 12
X intercept = (3,0)

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MATHS
Y intercept = (0,4)
6.
3y -4x = -12
X intercept = (3,0)
Y intercept = (0,-4)
7.
3y-4x-8=0
X intercept =(-2,0)
Y intercept = (0,8/3)

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MATHS
Part B:
formula relating height(H in meters) and boiling point of water above sea level B (in Celsius)
given by,
B + 0.0034H = 100
1. The independent variable is H thus this is the x value
2. The dependent variable is B thus this the y value
3. X intercept = (29411.76,0)
Y intercept = (0,100)
4. y intercept gives the boiling point of water at exactly sea level.

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MATHS
5. 0.0034 is the increment in Boiling point temperature in Celsius for a meter increase in
height.
Activity 11:
In the above figure it is somehow difficult to find the y intercepts as lines are crossing each
other. Hence, for finding the equation of the line a different technique can be applied. The
end points of the lines can easily be determined as they starts and ends with a whole number
of grid points. Thus after determining start and end point let (x1,y1) and (x2,y2) the equation
of a line can be determined by the following equation
(y-y1) = (x-x1)*(y2-y1)/(x2-x1) (point slope form of straight line)
Similarly using the above equation the equations of all the straight lines can be determined.
Activity 12:
Let the two other companies from which Chad could rent a car are Supercars and Car-times.
The breakdown of total costs of Supercars is $35 fixed initial charge and $0.50 per kilometre
charge.
The breakdown of total cost of Car-times is $40 fixed initial charge and $0.40 per kilometre
charge.

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MATHS
Flyer for Supercars:
Flyer for Car-times:
SuperCars
Rent a car at amazing price
Car-times

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MATHS
Let the independent variable is distance travelled by cars d kms.
Hence, equation of total cost of renting a car from Supercars will be
Ps = 35 + 0.5*d
Similarly, equation of total cost of renting a car from Car-times will be
Pc = 40 + 0.4*d
The price of both companies are represented by the following graph
Here, the red line is the total cost Ps and blue line is the total cost Pc.
The point that represents equal cost both companies can be found equating both equations
Ps = Pc
 35 + 0.5*d = 40 + 0.4*d
 0.1d = 5
 d = 50 kms.
Hence, when rented car travels for 50 kms exactly then the cost of both companies are equal.
The total cost at that point is 35 + 0.5*50 = $60.

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MATHS
Hence, for Chad it will benefitted to use car from Supercars if he travels less than 50 kms or
if he travels over 50 kms then it will be benefitted to use Car-times service and he can use
either one if he wants to travel exactly 50 kms with same cost.
Activity 13:
Standard units sold s = 5
Deluxe unit sold d= 4
Total selling price = $39000
In next month standard units sold s = 4
Deluxe unit sold d = 2
Total selling price = $24000
1. Equation to represent first month’s sales
5s + 4d = 39000 (1)
2. Equation to represent second month’s sales
4s + 2d = 24000 (2)
Now, solving (1) and (2) by substitution method
4d = 39000 – 5s => 2d = (39000 – 5s)/2
Substituting in (2) gives
4s + (39000 – 5s)/2 = 24000
 8s + 39000 – 5s = 48000
 3s = 9000
 s = $3000

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MATHS
Hence, 2d = (39000-5*3000)/2 = 12000
 d = $6000
3. Hence, standard units are sold for $3000
4. Deluxe units are sold for $6000
Activity 14:
1. Let hourly rate = $h per hour and overtime rate = $o per hour.
Hence, for first week total 14.5 hours worked.
Hence, 10*h + 4.5*o = 93.75 (1)
In the second week total 12 hours worked.
Hence, 10*h + 2*o = 75 (2)
Thus, 10h = 93.75 – 4.5o from (1)
And, 10h = 75 – 2o from (2)
Hence, 93.75 – 4.5o = 75-2o
 2.5o = 18.75
 o = $7.5 per hour
Hence, h = (93.75 – 4.5*7.5)/10 = $6 per hour.
Hence, hourly wage of Cathy is $6 per hour and overtime wage of Cathy is $7.5 per hour.
2.
Let jeans price = j
Blouse price = b

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MATHS
Hence, by Cathy’s bill
3b + 2j = 57 (1)
And, by Brenda’s bill
4b + 3j = 81 (2)
Multiplying (1) by 3 and (2) by 2 and then subtracting (1) from (2) gives
9b + 6j = 171
-8b - 6j = -162
=> b = $9
Hence, j = (57-3*9)/2 = $15
Hence, price of a blouse is $9 and price of a pair of jeans is $15.
3.
Let cost of hamburger = $h
Let cost of French fries = $f
Hence, the system of linear equation will be
8h + 5f = 24 (1)
6h + 2f = 16.60 (2)
Multiplying (1) by 2 and (2) by 5 and then subtracting gives
16h + 10f = 48
-30h - 10f = -83
=> -14h = -35

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MATHS
=> h = $2.5
Hence, f = (24-8*2.5)/5 = $0.8
Hence, price of a hamburger is $2.5 and price of a French fry is $0.8.
Activity 15:
Given, initial cost of manufacture = $20000
Per unit game manufacture cost after initial cost = $4 per unit
Wholesale price = $15 per game
Part A:
1.
The total cost of manufacturing any number of games is equal to $20000 added with per unit
cost multiplied by number of units.
2. Rewriting equation with x as independent and y as dependent variable
y = 20000+ 4x (1)
y = total cost of manufacturing
x = number of units manufactured
3. The revenue of any number of games sold is $15 multiplied with number of games sold.
4. Rewriting equation with x as independent and y as dependent variable
y = 15x (2)
y = revenue
x = number of units sold

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MATHS
5. Now, both equations are entered into graphing calculator.
6. Now, the point of intersection is found by trace
7.
The exact point of intersection can’t be found. The best estimate is (1818.182, 27272.727)
8.
Now, zoom function is used to zoom closer to the point of intersection.

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MATHS
Now, zooming the plot also gives the same number of decimal points by graphing calculator.
The best estimated point of intersection is (1818.182, 27272.727).
9.
The point of intersection between the revenue line and the price line is known as the break-
even point. At this point no profit or loss is incurred by the manufacturer. This point is
important because this gives an estimate of how many products needs to be sold to make
profit or not to incur loss.
The number of games that are needed to be sold before making profit is obtained from the
graph as 1818.
This number is slightly different from the x value of point of intersection as the exact solution
is in decimals but games cannot be sold in fractions. Thus the number of games is
approximated to nearest integer.

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MATHS
Part B:
1. Now, the exact value of point of is found using substitution method.
By putting y of equation (2) in (1) gives
15x = 20000 + 4x
 11x = 20000
 x = 20000/11
Hence, exactly 20000/11 games must be sold before expecting some profit.
2.
The revenue at this point will be
y = 15*20000/11 = $300000/11
3.
The cost at this point is
y = 20000+ 4*20000/11 = (220000 + 80000)/11 = $300000/11
Hence, cost = revenue or the point is break even.
4. The profit at this point = Revenue – cost = $0
5. The revenue for selling 2000 games = 15*2000 = $30000
Cost of selling 2000 games = 20000 + 4*2000 = $28000
Hence, profit of selling 2000 games = $30000 - $28000 = $2000
The revenue for selling 5000 games = 15*5000 = $75000
Cost of selling 5000 games = 20000 + 4*5000 = $40000

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MATHS
The profit for selling 5000 games = $75000 - $40000 = $35000
Revenue of selling 10000 games = 15*10000= $150000
Cost of selling 10000 games = 20000 + 4*10000 = $60000
The profit for selling 10000 games = $150000 - $60000 = $90000
Creating linear equation:
1.
Graph plot of y vs x:
0 1 2 3 4 5 6
0
2
4
6
8
10
12
14
16
18
20
Line plot
x (number of graphics)
y ($)
2.
From the above graph the initial value or the y intercept is (0,6)
3.
Slope = (y2 – y1)/(x2-x1)
Where (x1,y1) and (x2,y2) are any two points on the line.
Considering first two points Slope = (8.5-6)/(1-0) = 2.5

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MATHS
Hence, the rate per litre is $2.5 per litre.
4. The final cost of t-shirt with number of graphics printed x is given by,
y = mx + c (where m = slope and c = y intercept)
= 6 + 2.5x
5.
If the cost per graphic decrease by $2.25 then the y intercept of the equation will reduce by
2.25.
Hence, the new equation will be
y = 3.75 + 2.5x
6.
Let the total price of tea shirt is 35.5 for graphic of 10 and rate of $2.75 then
35.5 = c + 2.75*10
=> c = 35.5 – 27.5 = 8
Hence, the new equation will be
y = 8 + 2.75x
7.
In the new case the initial value is (0,8).
8.
The initial value represents price is $8 for zero graphic number.
9.

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MATHS
The graph of new equation y = 8 + 2.75x is given below.
0 1 2 3 4 5 6
0
5
10
15
20
25
New equation graph
x (number of graphics)
y ($)