How to Solve Linear Equations?

We can use linear equations in our day-to-day lives, for example, when comparing rates, making predictions, budgeting, and more. An equation is said to be linear when all the variables have the highest power equal to one. The variables are dependent on one another. This type of equation is also called a one-degree equation. Read on to discover the linear equation standard form, formula, graph, and guidelines to solve a linear equation in one or two variables. 

Here is what we will cover in the article:

• What is a linear equation?
• Linear equation standard form
• Linear equation graph
• Linear equations in one variable
• Linear equations in two variables
• How do solve linear equations?

What is a Linear Equation?

An equation with the highest degree of 1 is linear. It has no variable with an exponent of more than 1. A linear equation always forms a straight line on a graph, and thus, it gets its name ‘linear equation’.

In Mathematics, we have linear equations in one variable and two variables. The following examples will help you learn how to differentiate linear equations from nonlinear ones. 

Equations	Linear or Nonlinear
y = 7x – 8	Linear
y = x3 – 6	Non-Linear as the power of the variable x is 3.
√y + x = 5	Non-Linear because the power of y is ½.
y + 5x – 3 = 0	Linear

Linear Equation Formula

The linear equation formula expresses a linear equation. There are different ways a linear equation can be expressed. For example, we have the standard form, the point-slope form, or the slope-intercept form.

Linear Equations in Standard Form

The standard form of linear equations in one variable can be given as follows:

Ax + B = 0

The terms in the above equation denote:

• A and B are real numbers
• x is the single variable

The standard form of linear equations in two variables can be given as follows:

Ax + By = C

The terms in the above equation denote:

• A B and C are real numbers.
• A is the coefficient of x
• B is the coefficient of y
• C is constant
• x and y are the variables

Linear Equation Graph

On graphing a linear equation in one variable x, we get a vertical line parallel to the y-axis. If we graph a linear equation in two variables, x, and y, it forms a straight line. To graph, a linear equation, follow the steps given below.

• Step 1: Note down your linear equation and convert it into the form of y = mx + b. 
• Step 2: When we have our equation in this form, we can replace the value of x with various numbers. We will get the resulting value of y, and we can create the coordinates.
• Step 3: Next, we will list the coordinates in tabular form. 
• Step 4: Finally, we will plot the coordinates on a graph. Join the points to get a straight line. This line will represent our linear equation. 

How to Solve Linear Equations?

The main goal of solving an equation is to balance the two sides. An equation resembles a weighing balance as we have to ensure that both sides weigh equally. So, if we add a number on one side, we must add it to the other side as well.

Similarly, if we divide or multiply a number on the left-hand side, we will do the same for the right-hand side. To solve a linear equation, we will bring the variables to one side and keep the constant on the other side. Then we will find the value of the unknown variable. 

Tips on Solving Linear Equations:

• The solution or root of the linear equation is the value of the variable. It makes a linear equation true.  
• This solution remains unaffected if the same number is multiplied, added, subtracted, or divided for both sides of the equation.
• On graphing a linear equation in one or two variables, we get a straight line. 

Solution of Linear Equations in One Variable

We need to create a balance on both sides of the linear equation to solve it. The equality sign symbolizes that the expressions are equal on the two sides. The following sample linear equation will help you understand the steps of solving the linear equation in one variable.

Example 1: Solve (2x – 4)/2 = 3(x – 1)

Step 1: We will clear the fraction

x – 2 = 3(x – 1)

Step 2: We will simplify the two sides of the equations by opening the brackets and multiplying the number by the inner terms of the bracket. 

x – 2 = 3x – 3

Step 3: Next, we will isolate x

3-2 = 3x – x

1 = 2x

½ = x

0.5 = x

Solution of Linear Equations in Two Variables

There are several methods for solving a linear equation with 2 variables. Some commonly used methods are: 

• Method of substitution
• Cross multiplication method
• Method of elimination

Substitution Method

We use the substitution method when we have two linear equations with two unknown values. The following steps will help you solve the linear equations. 

• Step 1: First, simplify the given equation. Expand the parenthesis.
• Step 2: Now, we will solve one of the equations to obtain the value of either x or y.
• Step 3: Substitute the value of x in terms of y in the other equation or the value of y in terms of x.  
• Step 4: Solve the new equation following the basic arithmetic operations rules (BODMAS/ DMAS) to find the value of a variable.
• Step 5: Finally, use the value obtained and find the value of the second variable.

Example 2: Calculate the value of x and y from the equations 2x+3y = 13 and x-2y = -4

2x + 3y = 13 —————— (i)

x-2y = -4 ———————– (ii)

Solving eq (ii)

x = 2y -4

Substituting the value of x in eq ( i)

2 ( 2y -4) + 3y = 13

4y -8 + 3y =13

7y = 13+8

y= 21/7

y= 3

Substituting the value of y in eq (ii)

x= 2 (3) -4

x= 6-4

x=2

Cross Multiplication Method 

Cross multiplication is one of the simplest methods, and it is applicable only when we are given a pair of linear equations in two variables. We multiply the numerator of one fraction to the denominator of the other. The denominator of the first term is multiplied by the numerator of another term. The following equation is for solving linear equations in two variables using the cross-multiplication method. 

Elimination Method

The following steps will help you solve a linear equation:

• Step 1: First, we will multiply the given equations with non-zero constants. This process makes the coefficients of any one of the variables numerically equal.
• Step 2: Next, we will add or subtract one equation from the other. This step will eliminate one variable. Now, we will get an equation in one variable. 
• Step 3: We can now solve the equation in one variable and get its value.
• Step 4: Once we have the value of a variable, we can substitute this value in any one of the equations to calculate the other variable.

Example 3: Solve the following equations for x and y

2x+3y=6 —————— (i)

-2x+5y=10 —————-(ii)

We will add the two equations as follows

2x + 3y -2x +5y =6 +10

Since the coefficients of x are equal and opposite in sign, they will be eliminated.

8y = 16

y= 16/8

y= 2

Now, we will substitute the value of y in eq (i)

2x + 3(2) =6

2x + 6 = 6

 2x =0

x= 0

Example 4: If the difference in the measures of the given two complementary angles is 22°. Find the measure of the two angles. 

Solution: Let the angle be x. The complement of x = 90 – x

Given their difference = 22°

Therefore, (90 – x) – x = 22°

⇒ 90 – 2x = 22

⇒ -2x = 22 – 90

⇒ -2x = -68

⇒ x = 68/2

⇒ x = 34

The complementary angle will be 90 -34 = 56

Answer: The two complementary angles are 56 and 34. 

Practice Problems

Practice the following problems on linear equations to ace these questions in your examinations. 

Question 1: Solve the following linear equations using the substitution method.

1. 4x-3y=20 and 16x-6y=80
2. 2x-5y=10 and 3x+8y=15

Question 2: Solve linear equations given below using the elimination method.

1. 3x + y = 6 and 2x + 7y = 10.
2. 4x + 2y = 5 and 4x + 6y =15

Question 3: The sum of two numbers is 55. Suppose one number exceeds the other by 8. Find the two numbers.

Question 4: The length of a rectangle is thrice its breadth. If the perimeter of the rectangle is 32 meters, find the length and breadth of the rectangle.

Question 5: James is five years younger than Lily. Four years later, Lily will be twice as old as James. What is their present age of James? 

Question 6: The cost of three tables and two chairs is $605. If the table costs $50 more than the chair, what are the costs of the table and the chair?

Frequently Asked Questions

1. What is a Linear Equation?

Ans. A linear equation is an equation that has a straight line as its graph.

2. What are the Methods of Solving Linear Equations?

Ans. methods of solving linear equations are:

1. Addition and Subtraction
2. Multiplication and Division

3. How to Solve Linear Equations with Fractions?

Ans. Addition, Subtraction, Multiplication, and Division

3 Step 1: Find the slope.

Step 2: Find the y-intercept.

Step 3: Use these two numbers to find any missing values.

4. What is the formula for solving linear equations?

Ans. Linear equations are equations that have variables on both sides, and the variables are all multiplied by a constant. We can solve them by dividing both sides by the constant. For example, if you have an equation with 2x + 3 = 5 then you can solve it by dividing both sides by 2. This will give you x + 3/2 = 5/2 and then dividing both sides again gives you x = 5/4.

5. How do you calculate a linear equation?

Ans. To calculate a linear equation, you need to know the slope and y-intercept of the line. The slope is the number that tells you how much the line changes as it moves across the graph. The y-intercept is the point at which a horizontal line intersects with your graph. You can calculate these values by using the formula y=m*x+b, where m is your slope and b is your y-intercept.