## I. Introduction

Linear equations are one of the fundamental concepts in algebra that students encounter in middle school or high school. It involves a basic equation, where the variable is raised to the power of one and consists of linear terms. Solving linear equations is an essential skill for anyone who wants to pursue higher math, sciences or engineering.

This article provides a comprehensive guide on solving linear equations. We will cover all the necessary steps, common mistakes to avoid, and different approaches to solving linear equations. Furthermore, we will delve into the real-world applications of linear equations and advanced applications relevant to university-level math.

## II. Step-by-Step Guide to Solving Linear Equations

Before we dive into the step-by-step guide let’s first define what we are trying to solve for, the linear equation. A linear equation takes on the general form of *ax+b=c*, where x is the variable we are solving for, a, b, and c are constants. We may have multiple variables, but the focus would remain on obtaining the solution for the given variable.

The following are the steps to solving linear equations:

- Isolate the variable on one side of the equation.
- Use inverse operations to solve for the variable.
- Verify the solution by checking the answer.

The first step is to place all variable terms on one side of the equation, leaving constant terms on the other side. The second step is to perform inverse operations, which means undoing any arithmetic operation (addition, subtraction, multiplication, or division) to isolate the variable. Lastly, we assume the value obtained as the solution (x) and check our answers by substituting x back into the original equation.

Here’s an example of how to solve a linear equation:

2x+5=15

First, isolate the variable(2x) and move the constant to a separate side by subtracting 5 from both ends.

2x+5-5=15-5

2x=10

Next, divide by 2 on both sides to get the value of x.

2x/2 = 10/2

x=5

Lastly, check the answer by plugging x=5 back into the original equation:

2(5)+5=15

The result satisfies the original equation; hence, x=5 is the correct solution.

## III. Common Mistakes to Avoid When Solving Linear Equations

Common mistakes when solving linear equations include errors in simplifying or adding/subtracting values, distributing values inconsistently, or even changing the wrong terms.

It is vital always to remember the order of operations (PEMDAS) and avoid rushing through the problem to ensure no details are missed in the calculation. Double-checking the final answer before submission is another useful tip to detect any potential errors.

## IV. Solving Linear Equations Using Graphs

Another method to solve linear equations is by graphing them. Graphs are useful when dealing with complex expressions since they provide a visual representation of the problem. To graph a linear equation, we must know either the slope-intercept or point-slope form of the equation given.

Here is an example of graphing equation 3x+2y=6 using the slope-intercept form:

- First, we need to rearrange the equation and solve for y (the slope-intercept form)
- 2y=-3x+6
- y=-3/2x+3

The slope-intercept form is *y=mx+b *, where m is equal to the slope, and b is the y-intercept. From the given equation, we know that the slope (m) is -3/2, and the y-intercept (b) is 3.

We can now draw a line on our x-y graph using the slope (-3/2) and y-intercept (3) as a reference point. Next, we find the solution of the equation by locating the point (x,y) where the line intercepts the x-axis.

## V. Real-World Applications of Linear Equations

Linear equations have essential practical applications in various fields, such as ‘cost of good sold’ analysis in accounting, calculating depreciation of assets, and calculating rates of change in sciences.

One example of a real-world application of linear equations in everyday life is discounts given by retailers to attract customers. For instance, a clothing store provides 25% off on all products. A customer comes into the shop and buys a $200 jacket. The amount of discount she receives is the product price multiplied by the discount percentage, i.e.,

Discount = Price x (Discount Percentage/100)

Discount= $200 x (25/100)

Discount= $50

Therefore, the final cost of the jacket the customer pays is $200 – $50 = $150.

## VI. Different Strategies for Solving Linear Equations

There are different strategies to solve linear equations, including substitution, elimination, matrix, and graphical methods.

The substitution method involves isolating one variable in a linear equation and substituting the result into the other equation to obtain an equation in one variable. The elimination method requires adding or subtracting equations to remove one variable simultaneously. Matrix methods involve Gauss-Jordan elimination and finding the inverse of coefficients of variables. Graphical methods involve plotting linear equations on an x-y plane and determining the point of intersection as a solution.

One example of applying substitution is solving a system of equations. Say, for instance, we have two linear equations:

2x + 3y = 7

3x + 2y = 8

From the first equation place 2x=7-3y and substitute in the second equation.

3(7-3y/2) + 2y = 8

21 – 9y/2 + 2y = 8

21-9y+4y=16

21-5y=16

-5y=-5

y=1

Substitute 1 in the first equation

2x+3(1)=7

2x=4

x=2

Thus the solution to the system of equations is (2,1).

## VII. Advanced Applications of Linear Equations

Advanced applications of linear equations include solving systems of differential equations in physics, linear programming in economics, and machine learning in computer science. Matrices are essential to solving systems of linear equations in matrix algebra.

Example:

Consider the following system of equations:

x + y + z = 6

2y + 5z = -4

2x + 5y – z = 27

We can create a matrix system for this with one matrix containing all the coefficients and a separate one having the values we are setting equal to:

[1 1 1 [6

0 2 5 -4

2 5 -1] =27]

Next, we could solve for the identity matrix by using the Row Echelon form:

[1 1 1 6 [1 1 1 6

0 2 5 -4 –> 0 2 5 -4/2 =-2

0 3 -3 15] 0 1 1 5

Finally, we would use back-substitution to solve for the variables: x=1, y=2, z=3.

## VIII. Conclusion

Solving linear equations is a fundamental concept that we use in many practical areas of life. It can be challenging and intimidating for many students, but with perseverance, anyone can master it. In this article, we have covered all the necessary details, including the step-by-step guide, common mistakes, different strategies, and advanced applications of linear equations. We encourage readers to practice and apply these concepts to excel in their math and science courses.