Linear algebra is fundamental in computer science, underpinning algorithms, graphics, machine learning, and data analysis. Understanding linear equations and how to solve them is essential for tasks such as modeling networks, optimizing computations, and handling multidimensional data. This chapter introduces linear equations and demonstrates systematic methods for solving them, equipping learners with practical skills for computational problem-solving in Kenyan tech industries and research environments.

1.1 Linear Equations

Linear equations express relationships where variables appear to the first power only, forming straight-line graphs in two dimensions. They are widely used in computer science for system modeling, cryptography, and algorithm design. The general form of a linear equation in one variable is

$$ ax + b = 0 $$

where \(a\) and \(b\) are constants and \(x\) is the variable to solve for.

1.1.1 Linear Equations in One Variable

Linear equations in one variable involve expressions with a single unknown raised to the first power. The goal is to isolate the variable to find its value.

$$ ax + b = 0 $$

where \(a \neq 0\).

Worked Examples

Example 1: A software developer is debugging a program where the execution time \(t\) (in seconds) depends linearly on the number of instructions \(n\). Given the equation \(5t + 20 = 0\), find \(t\).

Given: \(a=5\), \(b=20\)

$$ 5t + 20 = 0 $$

$$ 5t = -20 $$

$$ t = \frac{-20}{5} $$

Answer: \(t = -4\) seconds

Example 2: In a database query, the number of rows processed \(r\) satisfies \(3r - 12 = 0\). Find \(r\).

Given: \(a=3\), \(b=-12\)

$$ 3r - 12 = 0 $$

$$ 3r = 12 $$

$$ r = \frac{12}{3} $$

Answer: \(r = 4\) rows

Example 3: A cybersecurity model uses the equation \(7x + 14 = 0\) to represent a threshold condition. Find \(x\).

Given: \(a=7\), \(b=14\)

$$ 7x + 14 = 0 $$

$$ 7x = -14 $$

$$ x = \frac{-14}{7} $$

Answer: \(x = -2\)

1.1.2 Linear Equations in Two Variables

Linear equations in two variables represent lines on a plane and are generally written as

$$ ax + by = c $$

where \(a\), \(b\), and \(c\) are constants, and \(x\), \(y\) are variables.

Worked Examples

Example 1: A data scientist is analyzing the relationship between hours studied \(x\) and exam score \(y\) given by \(2x + 3y = 12\). Find \(y\) when \(x=3\).

Given: \(a=2\), \(b=3\), \(c=12\), \(x=3\)

$$ 2(3) + 3y = 12 $$

$$ 6 + 3y = 12 $$

$$ 3y = 12 - 6 $$

$$ 3y = 6 $$

$$ y = \frac{6}{3} $$

Answer: \(y = 2\)

Example 2: In network traffic modeling, the equation \(4x - 5y = 20\) relates packets sent \(x\) and packets lost \(y\). Find \(y\) when \(x=10\).

Given: \(a=4\), \(b=-5\), \(c=20\), \(x=10\)

$$ 4(10) - 5y = 20 $$

$$ 40 - 5y = 20 $$

$$ -5y = 20 - 40 $$

$$ -5y = -20 $$

$$ y = \frac{-20}{-5} $$

Answer: \(y = 4\)

Example 3: A cloud service provider models resource allocation with \(3x + 2y = 18\). Find \(y\) when \(x=4\).

Given: \(a=3\), \(b=2\), \(c=18\), \(x=4\)

$$ 3(4) + 2y = 18 $$

$$ 12 + 2y = 18 $$

$$ 2y = 18 - 12 $$

$$ 2y = 6 $$

$$ y = \frac{6}{2} $$

Answer: \(y = 3\)

1.1.3 Systems of Linear Equations

Systems involve multiple linear equations with multiple variables, commonly solved to find variable values satisfying all equations simultaneously. In computer science, they appear in graphics transformations, error correction, and optimization.

For two variables, the system is

$$ \begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases} $$

Worked Examples

Example 1: A developer models CPU and memory usage with

$$ \begin{cases} x + y = 10 \\ 2x - y = 3 \end{cases} $$

Find \(x\) and \(y\).

Given: \(a_1=1, b_1=1, c_1=10; a_2=2, b_2=-1, c_2=3\)

Multiply the first equation by 1 and add to the second:

$$ x + y = 10 $$

$$ 2x - y = 3 $$

Add equations:

$$ (x + y) + (2x - y) = 10 + 3 $$

$$ 3x = 13 $$

$$ x = \frac{13}{3} $$

Substitute \(x\) into first equation:

$$ \frac{13}{3} + y = 10 $$

$$ y = 10 - \frac{13}{3} $$

$$ y = \frac{30}{3} - \frac{13}{3} $$

$$ y = \frac{17}{3} $$

Answer: \(x = \frac{13}{3}, y = \frac{17}{3}\)

Example 2: In database query optimization,

$$ \begin{cases} 3x + 4y = 24 \\ 5x - 2y = 10 \end{cases} $$

Find \(x\) and \(y\).

Given: \(a_1=3, b_1=4, c_1=24; a_2=5, b_2=-2, c_2=10\)

Multiply first equation by 2:

$$ 6x + 8y = 48 $$

Multiply second by 4:

$$ 20x - 8y = 40 $$

Add:

$$ 6x + 8y + 20x - 8y = 48 + 40 $$

$$ 26x = 88 $$

$$ x = \frac{88}{26} = \frac{44}{13} $$

Substitute into first equation:

$$ 3 \times \frac{44}{13} + 4y = 24 $$

$$ \frac{132}{13} + 4y = 24 $$

$$ 4y = 24 - \frac{132}{13} = \frac{312}{13} - \frac{132}{13} = \frac{180}{13} $$

$$ y = \frac{180}{52} = \frac{45}{13} $$

Answer: \(x = \frac{44}{13}, y = \frac{45}{13}\)

Example 3: A cloud computing model:

$$ \begin{cases} x - y = 5 \\ 3x + 2y = 16 \end{cases} $$

Find \(x\) and \(y\).

Given: \(a_1=1, b_1=-1, c_1=5; a_2=3, b_2=2, c_2=16\)

From first:

$$ x = y + 5 $$

Substitute in second:

$$ 3(y + 5) + 2y = 16 $$

$$ 3y + 15 + 2y = 16 $$

$$ 5y + 15 = 16 $$

$$ 5y = 1 $$

$$ y = \frac{1}{5} $$

Then

$$ x = \frac{1}{5} + 5 = \frac{1}{5} + \frac{25}{5} = \frac{26}{5} $$

Answer: \(x = \frac{26}{5}, y = \frac{1}{5}\)

Chapter Summary

This chapter covered the fundamentals of linear algebra starting with linear equations and their significance in solving mathematical problems. Various methods for solving linear equations were explored, including the formation of equations from real-world scenarios. The concept of vectors was introduced, defining their nature and different types encountered in computations. Vector operations such as addition, subtraction, and multiplication were discussed in detail, highlighting scalar multiplication and the dot product as essential tools. Matrices were defined along with their types, the calculation of determinants, and their applications in computer science were explained. Operations on matrices including addition, scalar multiplication, and transposition were demonstrated to show how matrices can be manipulated. Finally, the chapter explained the inverse of a square matrix and its importance in solving matrix equations. This comprehensive overview provides the foundation needed for advanced studies in linear algebra within computer science.