Solving Linear Equations: Formulas, Methods, and Questions

Linear equations can be understood as algebraic equations having an order of unity. In mathematics, an equation is a mathematical representation of real-world issues in the form of analogous connections expressed as variables and constants. Algebraic equations are made up of algebraic expressions on both sides of the equal symbol (=). There are various methods of solving linear equations such as the substitution method, elimination method, the matrix method, Cramer’s rule, etc. In this article, we will look at the various methods of solving linear equations.

Read More:- Linear Equation in Two Variable

Key Takeaways: Linear Equation, Algebraic Equations, Cramer’s Rule, Determinants, Elimination Method, Substitution Method, Matrix

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What is Linear Equation?

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A linear equation is an algebraic equation in which the variable has the largest exponent of one. There are one, two, or three variables in a linear equation, however not every linear system has 03 equations. An algebraic expression and an equal to (=) sign are combined to form a linear equation. It is known as a first-degree equation since it has a degree of one. For example, x + y = 4. A linear equation's solution is simply the value or values of the variables involved in the equation. Finding the solution to linear equations in one, two, three, or more variables is known as solving a linear equation.

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Types of Linear Equation

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A system of linear equations usually have just one solution, but it can also have no solution or an unlimited number of solutions. A two-variable linear equation is one in which the value of one variable, say 'x,' is dependent on the value of the second variable, say 'y.' A linear equation's graph will be a straight line if there are two variables.

Based on the number of variables in the equation, linear equations are divided into distinct categories. The following are the many types of linear equations:

• Linear equations with only one variable, like ax + b = 0, are known as one-variable linear equations.

Example: 12x – 10 = 0

• Linear equations having two variables, such as ax + by + c = 0, are two-variable linear equations.

Example: 12x +10y – 10 = 0

12x +23y = 20

• Three-variable linear equations are the linear equations that have three variables like ax+ by + cz + d = 0.

Example: 12x +10y - 3z – 10 = 0

12x +23y – 12z = 20

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Methods of Solving Linear Equations

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The various methods of solving linear equations are:

1. Substitution Method for Solving Linear Equations

In the substitution method, we change the equation such that one of the values is substituted in the second equation. Now that we only have one variable in our equation, we can solve it and get the value of that variable. Any equation may be taken from the two given equations, and the value of a variable can be obtained and substituted in another equation.

Example: x + y = 6 —(1),    2x + 4y = 20 — (2)

Step 1: Using any of the equations, get the value of one of the variables.

x + y = 6 — (1)

x = 6 – y

Step 2: In the second linear equation, substitute the value of the variable determined in step 1.

2(6 - y) + 4y = 20

12 - 2y + 4y = 20

12 + 2y = 20

2y = 20 - 12

2y = 8

y = 8/2

y = 4

Step 3: Now, in either equation (1) or (2), substitute the value of 'y'.

x + y = 6

x + y = 6

x = 6 - 4

x = 2

The linear equations are therefore solved using the substitution approach, and the values of x and y are 2 and 4, respectively.

2. Solving Linear Equations by Elimination Method

In the elimination method, we try to multiply either the 'x' variable term or the 'y' variable term with a constant value so that the 'x' variable terms or the 'y' variable terms cancel out and the other variable's value is obtained. Let's look at how to solve linear equations using the elimination approach.

Consider the following equations.

2x + 3y = 9 — (i)

x – y = 3 — (ii)

When you multiply equation (ii) by 2, the coefficient of "x" becomes the same and may be removed.

As a result, multiply equation (ii) by2 and then remove it from equation (i).

2x + 3y = 9

(-) 2x – 2y = 6

___________

-5y   =  -3

Or, y =3/5 = 0.6

Put the value of y = 0.6 in the equation now (ii).

So, x – 0.6 = 3

Thus, x = 3.6

3. Solving Linear Equations by Cross Multiplication

By selecting the coefficients of all the terms ('x', 'y', and the constant terms) in the manner indicated below and applying the procedure for obtaining the values of 'x' and 'y', we may solve linear equations.

Solving Linear Equations by Cross Multiplication

Formula = \(\frac{x}{b_1c_2-b_2c_1} = \frac{y}{c_1a_2-c_2a_1} = \frac{1}{a_1b_2-a_2b_1}\)

\(\frac{x}{b_1c_2-b_2c_1} = \frac{1}{a_1b_2-a_2b_1}\)

\(\frac{y}{c_1a_2-c_2a_1} = \frac{1}{a_1b_2-a_2b_1}\)

Value of x = \(\frac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1}\)

and y= \(\frac{c_1a_2-c_1a_1}{a_1b_2-a_2b_1}\)

4. Graphical Method for Solving Linear Equations

The graph is another approach for solving linear equations. We graph both equations when given a system of linear equations by obtaining values for 'y' for different values of 'x' in the coordinate system. After that, we locate the location where these two lines cross. The solution for these linear equations is determined by the (x,y) variables at the point of intersection. Let's look at two linear equations and see how the graphical technique can help us solve them.

x + y = 8…...(1)

y = x + 2…...(2)

Let's look at some 'x' values and see what the 'y' values are for the equation:

x + y = 8

y = 8- x

Let's look at some 'x' numbers and see what the values are for 'y' in the equation y = x + 2.

This graph is created by plotting these points on the coordinate plane.

To obtain the values of 'x' and 'y,' we must first identify the point of intersection of these lines. At the place where the two lines meet (3,5). As a result, using the graphical approach of solving linear equations, x = 3 and y = 5.

5. Matrix Method of Solving Linear Equation

The matrix technique is highly useful for solving two- or three-variable linear equations. Consider the following three equations:

a₁x + a₂y + a₃z = d₁ ; b₁x + b₂y + b₃z = d₂ ; c₁x + c₂y + c₃z = d₃

These equations are written as follows:

[a₁x + a₂y + a₃z '; b₁x + b₂y + b₃z ; c₁x + c₂y + c₃z ] = d₁ d₂ d₃

[a₁ a₂ a₃ b₁ b₂ b₃ c₁ c₂ c₃ x y z ] = d₁ d₂ d₃ =  AX = B-------(i)

The A, B, and X matrices are as follows:

A= [a₁ a₂ a₃ b₁ b₂ b₃ c₁ c₂ c₃] , X= [x y z] ,B = [d₁ d₂ d₃]

multiplying (i) by A^-1 to obtain:

A^-1AX = A^-1B

⇒ I.X = A^-1B

⇒ X = A^-1B

6. Cramer's Rule (Determinant Method of Solving Linear Equations)

The determinants method is a simple way to solve linear equations with two or three variables. The technique is as follows for linear equations with two and three variables.

For Two-Variable Linear Equations:

x = Δ₁/Δ, y = Δ₂/Δ

0r, x = (b₁c₂ – b₂c₁)/(b₂a₁ – b₁a₂) and y = (c₁a₂ – c₂a₁)/(b₂a₁ – b₁a₂)

\(\Delta_1 = \begin{bmatrix}b_1 & c_1 \\[0.3em]b_2 & c_2 \\[0.3em] \end{bmatrix}\), \(\Delta_2 = \begin{bmatrix}c_1 & a_1 \\[0.3em]c_2 & a_2 \\[0.3em] \end{bmatrix}\)and \(\Delta = \begin{bmatrix}a_1 & b_1 \\[0.3em]a_2 & b_2 \\[0.3em] \end{bmatrix}\)

In the case of three-variable linear equations:

\(\Delta = \begin{bmatrix}a_1 & b_1 & c_1\\[0.3em]a_2 & b_2 & c_2\\[0.3em]a_3 & b_3 & c_3\\[0.3em] \end{bmatrix}\), \(\Delta_1 = \begin{bmatrix}d_1 & b_1 & c_1\\[0.3em]d_2 & b_2 & c_2\\[0.3em]d_3 & b_3 & c_3\\[0.3em] \end{bmatrix}\), \(\Delta_2 = \begin{bmatrix}a_1 & d_1 & c_1\\[0.3em]a_2 & d_2 & c_2\\[0.3em]a_3 & d_3 & c_3\\[0.3em] \end{bmatrix}\), \(\Delta_3 = \begin{bmatrix}a_1 & b_1 & d_1\\[0.3em]a_2 & b_2 & d_2\\[0.3em]a_3 & b_3 & d_3\\[0.3em] \end{bmatrix}\)

7. Solving Linear Equation in One Variable

A one-variable linear equation is an equation with only one variable. To solve it, simplify the equation to bring the variable terms to one side and the constant value to the other side when solving linear equations with one variable. Find the LCM (Least Common Multiple) of any fractional terms and simplify them.

Example:

• 3x - 6 = 0

3x = 6

x = 6/3

x = 2

• 8x - 10 = 4x + 8

Let's simplify to determine the value of 'x' by putting the 'x' terms on one side and the constant terms on the other.

4x - 8x = -10 - 8

-4x = -18

4x = 18

x = 18/4

On simplifying, x = 9/2.

• Solving Linear Equation in Two Variables: Any of the above-mentioned methods can be used to solve linear equations in two variables
• Solving Linear Equation in Three Variables: Cross multiplication and matrix method is highly feasible for solving linear equation in three variables.

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Things to Remember

• Regardless of the method employed to solve the equations, the solution of a pair of linear equations examples in two variables stays the same.
• Determinant techniques can also be used to solve two-variable linear equations.
• A linear equation in two variables can be solved using any methods such as the graphical technique, elimination method, substitution method, cross multiplication method, matrix method, and determinants method.
• The graphical, elimination, and substitution methods cannot be used to solve any problem with three or more variables.
• The cross-multiplication approach is the most recommended method for solving a three-variable problem.
• Matrix Cramer's rule is also very helpful for solving problems with three or more variables.

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