Detailed Solutions for Simultaneous Linear Equations Problems

This assignment provides a comprehensive guide to solving simultaneous linear equations. It covers various methods including substitution and elimination, applied to different sets of equations. The solution demonstrates step-by-step processes, making complex problems easier to understand. The assignment includes solving pairs of equations using both substitution and elimination. It also involves solving equations with fractions and variables on both sides. The document concludes with a bibliography of relevant mathematical resources. This assignment is a valuable resource for students studying foundations of mathematics, offering clear explanations and practical examples to enhance understanding and problem-solving skills.

Surname 1
Student’s Name
Professor’s Name
Course Name
Date
Simultaneous Linear Equations
1. Solving the following pairs of simultaneous equations:
a)
x − 3y = 1
2x + 5y = 35
Using substitution method, the equations can be labeled as 1 and 2 respectively.
x − 3y = 1 (1)
2x + 5y = 35 (2)
For (1), x = 1 + 3y
Substitute this in equation 2 and solve for y
2(1+3y) +5(y) = 35
2 + 6y + 5y = 35
2+11y=35
11y=33
Y = 3
Substitute the value of y in equation 2 to solve for x
X - 3(3) = 1
X – 9 = 1

Surname 2
X = 1+9
x = 10
b)
2x + 1/3y = 1
3x + 5y = 6
Using elimination method, the two equations can be solved also.
2x + 1/3y = 1 (1)
3x + 5y = 6 (2)
Start by eliminating x by multiplying the two as follows:
3 2x + 1/3y = 1
2 3x + 5y = 6
6x + y = 3
6x + 10y = 12
-9y = -y
Y = 1
Also eliminate y by multiplying the two as follows
5 2x + 1/3y = 1
1/3 3x + 5y = 6
10x + 5/3y = 3
X + 5/3y = 2
Subtracting the two gives;
9x = 3
X = 1/3

Surname 3
c)
4x + 3y = 5
2x – 3/4y = 1
This can also be solved by elimination method as shown below:
2x + 1/3y = 1 (1)
3x + 5y = 6 (2)
Start by eliminating y by multiplying the two as follows:
-3/4 2x + 1/3y = 1
3 2x -3/4 y = 1
-3x - 9/4y = -15/4
6x- 9/4y = 3
-9x = -27/4
X = 3/4
Also eliminate x by multiplying the two as follows
2 4x + 3y = 5
4 2x – 3/4y = 1
8x + 6y = 10
8x - 3y = 4
Subtracting the two gives;
9y = 6
Y = 2/3

Surname 4
2. Using elimination to solve these following pairs of simultaneous equation
a)
1/3x + y = 10/3
2x + 1/4y = 11/4
Solve for x by eliminating y as shown below
1/4 1/3x + y = 10/3
1 2x + 1/4y = 11/4
1/12x+1/4y = 10/12
2x+1/4y = 11/4
Subtracting the two gives;
-23/12x = -23/12
X = 1
Again, solve for y by eliminating x as shown
2 1/3x + y = 10/3
1/3 2x + 1/4y = 11/4
2/3x + 2y = 20/3
2/3x + 1/12y =11/12
Subtracting the two gives;
-23/12y = -23/4
y = 23/4 * 12/23
y = 3
b)
3x − 2y = 5/2

Surname 5
1/3x + 3y = −4/3
Solve for y by eliminating x as shown below
1/3 3x - 2y = 5/2
3 1/3x + 3y = -4/3
X- 2/3y = 5/6
X + 9y = -12/3
Subtracting the two gives;
-29/3y = 29/6
y= -1/2
Again, solve for x by eliminating y as shown
3 3x -2 y = 5/2
-2 1/3x + 3y = -4/3
9x - 6y = 15/2
-2/3x - 6y = 8/3
Subtracting the two gives;
29/3x = 29/6
X = 29/6 * 3/29
x = 1/2
3. Solve these pairs of simultaneous equations by any method.
a)
x = 3y
4x − 5y = 35
Using substitution method, the equations can be labeled as 1 and 2 respectively.

Surname 6
x = 3y (1)
4x - 5y = 35 (2)
For (1), x =3y
Substitute this in equation 2 and solve for y
4(3y) -5(y) = 35
12y - 5y = 35
7y = 35
Y = 5
Solving for x
X = 3y
X = 3*5 =15
b)
x = 1/3y
2y − 6x = 9
Using substitution method, the equations can be labeled as 1 and 2 respectively.
x = 1/3y (1)
2y - 6x = 9 (2)
Substitute x into equation 2 and solve for y
2(y) - 6 (1/3y) = 9
2y - 1/2y = 9
3/2y = 9
Y = 9*2/3
Y = 6

Surname 7
Substitute the value of y in equation 2 to solve for x
2(6) - 6(x) = 9
-6x = 9-12
-6x = -3
X = 1/2
c)
7x + 3y = −15
12y − 5x = 39
Solve for y by eliminating x as shown below
-5 7x +3y = -15
7 -5x + 12y = 39
-35x - 15y = 75
-35x + 84y = 273
Subtracting the two gives;
-99y = -198
Y = 2
Again, solve for x by eliminating y as shown
12 7x +3 y = -15
3 -5x + 12y = 39
84x + 36y = -180
-15x + 36y = 117
Subtracting the two gives;
99x = -297

Surname 8
X = -3
3. Solve the equations
a) 5 + x/ 3 = 7
x/3 = 7-5
x/3 = 2
Hence, x = 2*3
X = 6
b)
3/4x−2 =1/3x + 3
Like terms together
3/4x −1/3x = 3+2
5/12x =5
X=5 * 12/5
X = 12

Surname 9
Bibliography
Milne, William Edmund. Numerical calculus. Princeton University Press, 2015.
Ostrowski, Alexander M. Solution of Equations and Systems of Equations: Pure and Applied
Mathematics: A Series of Monographs and Textbooks. Vol. 9. Elsevier, 2016.
Rosser, Mike, and Piotr Lis. Basic mathematics for economists. Routledge, 2016.
Searle, Shayle R., and Andre I. Khuri. Matrix algebra useful for statistics. John Wiley & Sons,
2017.