Solving Linear Equations

A linear equation is an equation where the variable’s exponent is exactly 1. For this reason, it’s also called the first-degree equation. When plotted, these equations create straight-line graphs. 

A linear equation in one variable looks like:
Ax + B = 0
where A and B are numbers, and x is the variable. 

A linear equation in two variables is written as:
Ax + By = C
where x and y are variables, A and B are coefficients, and C is a constant.
 

Solving a linear equation means finding the value of the variable that makes the equation true. These equations are called linear because they form a straight line when graphed. To solve the linear equation means to find the value of the variable that makes the equation true. This means finding the number that can replace x so that both sides of the equation are equal. The basic steps to solve linear equations are:

1. Simplify both sides by combining the like terms, or use the distributive property to remove parentheses.
2. Move the variable terms to one side and constants to the other side.
3. Isolate the variable and use multiplication or division to make the coefficient of the variable 1. 

We can solve linear equations that contain one, two, or three variables. A solution is simply the value of the variable that makes the equation true.

Solving Linear Equations in One Variable

A linear equation in one variable can be solved by isolating the variable step by step. The goal is to find the value that makes both sides of the equation equal. 
Example:
Solve: 3x + 2 = 11

Step 1: Subtract 2 from both sides to move the constant:
3x + 2 - 2 = 11 - 2
3x = 9

Step 2: Divide both sides by 3 to get x alone:
3x3 = 93
x = 3

Solving Linear Equations in Two Variables

x + y = 5 or 2x - y = 7 are some examples of linear equations in two variables. Since there are two unknown variables like x and y, we need two equations to solve them together. We can use any of the following methods to solve for linear equations in two variables:

• Graphical method
• Substitution method
• Elimination method
• Cross-multiplication and matrix methods

Graphical method: In graphical method, we need to draw both the equations as lines on the graph, the point where the lines meet is the solution.

Substitution method: Solve for one variable using one equation, and then substitute it into the other equation to find the value of the other variable.

Elimination method: Add or subtract the equations to eliminate one variable.

Cross-multiplication and matrix methods: These methods are efficient techniques for solving systems of linear equations with two or more variables.
 

When we have equations with three or more variables, we use special methods to get the best result possible:

• Cross-multiplication method
• Matrix method (using Cramer’s Rule)

These methods are helpful in solving equations quickly when there are many variables. 
 

There are many methods for solving linear equations. Some of the methods are:

• Graphical Method
• Elimination Method
• Substitution Method
• Cross Multiplication Method
• Matrix Method
• Determinants Method (Cramer’s Rule) 

Graphical Method

For solving linear equations in a graphical method, we draw the equations as a line on a graph. The point where the two lines cross or intersect is the answer.
Example:
y = 2x + 1
y = x + 3
Plot both lines on the graph.
Where they meet is the answer.
If they meet at (2, 5), that means x = 2 and y = 5 is the solution.

Elimination Method

In this method, we add or subtract two equations to cancel out one variable, allowing us to find the value of the remaining variable.
Example:
x + y = 10
x - y = 4
Add both equations,
(x + y) + (x - y) = 10 + 4
2x = 14
x = 7
Now put x = 7 in the first equation:
7 + y = 10
y = 3
So the final answer is x = 7, y = 3

Substitution Method

Substitution method means solving one equation to find one variable and then substituting it into the other.
Example:
Solve the system:
(1) y = x + 2
(2) x + y = 10
Step 1: Use equation (1) to substitute for y in equation (2):
x + (x + 2) = 10
Step 2: simplify the equation
2x + 2 = 10
2x = 8
x = 4
Step 3: Substitute x = 4 back into equation (1):
y = 4 + 2 = 6
Answer: x = 4, y = 6

Cross Multiplication Method

For solving linear equations using the cross multiplication method, we use formulas. The formula for solving two equations in the form of 
a1x + b1y = c1
a2x + b2y = c2
x = (b1c2 - b2c1)(a1b2 - a2b1), y = (c1a2 - c2a1)(a1b2 - a2b1)

Matrix Method

A matrix method is a neat way to write equations in rows and columns. Let's see this using an example.
x + y = 6
2x + 3y = 14
Write this in a matrix form as

Now use the formula:
AX = B
X = A-1B
Once we substitute the values and solve the equation, we get x = 4, y = 2.

Determinants Method (Cramer’s Rule)

This method uses determinants, which are specific numerical values calculated from square matrices and help in solving systems of equations.
Example:
2x + 3y = 12
x - y = 1
We find three determinants:
 from coefficients
1 replace the first column with answer numbers
2 replace second column with answer numbers
x = 1, y = 2

Linear equations are not only used in math, they have many real-world applications as well. Solving linear equations helps to find the unknown quantities using known relationships. Here are some of the real life applications of linear equations.

• Budgeting and finance: Linear equations are used to calculate monthly expenses or savings in budgeting and finance. If your monthly income is $2000, and you have to spend $x for rent and $800 on other expenses, then the equation is x + 800 = 2000. Solving this equation will give your rent. 

• Travel planning: In travel, linear equations are used to calculate distance, time, or speed. For example, if you are driving 60 km/h, and you need to reach 180 km, the equation is 60t = 180. Solving this gives time t.

• Cooking: Linear equations in cooking are used for adjusting ingredient amounts for serving sizes. If 4 cups of flour makes 8 pancakes, how much flour is needed for 20 pancakes. Use the equation (4/8)x = 20 or x = 20(4/8).