Linear Equation

A linear equation is an equation where the highest power of the variable is one. Therefore, a Linear Equation is also known as a one-degree equation. Read this article to learn more about the Linear Equations.

What is a Linear Equation?

A linear equation is an algebraic expression with the highest degree of any variable equal to 1. This means that no variable is raised to a power greater than 1. One must note that when it is plotted on a graph, a linear equation always produces a straight line.

Also Check: Circumstance of Circle

Types of Linear Equations

Linear equations can involve one or two variables. Definitions and examples of these types are discussed below.

Linear Equations in One Variable

The linear equation formed with a single variable forms the first type of Linear Equation. For an equation with one variable, the standard form is Ax + B = 0, where,

x represents the variable.

A is the coefficient of x.

B is a constant.

For example: x+2=5

Linear Equations in Two Variables

The linear equation formed with two variables form the second type of Linear Equation. For an equation with two variables, the standard form of a linear equation is Ax + By = C , where,

x and y are the variables.

A and B are coefficients.

C is a constant.

For example: x+2y=15

Also Check: Average

Linear Equations Examples

For a better understanding of one or two variables linear equations and non-linear equations, observe the table below.

Equations	Type of equations
x+2=15	One Variable Linear Equation
x+9y=10	Two-Variable Linear Equation
x+2z2 =35	Non-Linear Equation
x7 + 3y =5	Non-Linear Equation
4 +2y=18	One Variable Linear Equation

Linear Equation Forms

Linear equations can be expressed in several forms, including:

• Standard Form: Ax+By=C
• Slope-Intercept Form: y=mx+c

Here, m represents the slope of the line, and c is the y-intercept.

• Point-Slope Form: y – y1 = m(x – x1 )

In this form, m is the slope and (x1, y1) is a specific point on the line.

Methods to Solve the Linear Equations

Methods for solving the linear equations are discussed as follows:

Graphical Method: Plot the given linear equations on a graph to find their point of intersection.

Substitution Method: Solve one equation for one variable and substitute that into the other equation. This will give you the required solution.

Elimination Method: Add or subtract the equations to eliminate one variable, making it easier to solve for the other.

Cross-Multiplication Method: The cross-multiplication method is also a useful method for solving equations in the form of fractions.

Determinant Method (Cramer’s Rule): Determinant Method (Cramer’s Rule) uses determinants to solve the system of equations.

Also Check: CUBE

General Steps to Solve Linear Equations

• Start with simplifying Both Sides. You may combine like terms on both sides of the equation if needed.
• Isolate the Variable.
• Now move the variable terms to one side of the equation and constant terms to the other side.
• If the variable has a coefficient, divide both sides by this coefficient to solve for the variable.
• Substitute the solution back into the original equation to verify that it satisfies the equation.

Sample Example: Solve 10y – 5 = 25

Solution: 10y – 5 = 25

10y – 5 + 5 = 25 + 5

10y = 30

y = 3

Tips on Linear Equations

• The value of the variable that makes the equation true is called the solution or root.
• The equation remains balanced if the same number is added, subtracted, multiplied, or divided on both sides.
• The graph of a linear equation in one or two variables always forms a straight line.

Solved Examples of Linear Equations

Example 1: Solve 5x + 3 = 13

Ans. Subtract 3 from both sides:

5x + 3 – 3 = 13 – 3

Simplifies to:

5x=10

Divide both sides by 5:

x=2

Example 2: Solve 2y−7=3y+5

Ans. Subtract 2y from both sides:

2y−7−2y=3y+5−2y

Simplifies to:

−7=y+5

Subtract 5 from both sides:

−7−5=y+5−5

Simplifies to:

−12=y

y = -12

Example 3: Solve 2z − 5=3z +5

Ans. Subtract 2z from both sides:

2z−5−2z=3z+5−2z

Simplifies to:

−5 =z+5

Subtract 5 from both sides:

−5−5=z+5−5

Simplifies to:

−10=z

z = -10

Practice Questions on Linear Equations

1. 1. Solve the equation: 7x−4=10
2. 2. Solve the equation: 3(y+2)=18
3. 3. Solve the equation: 2x+4=3x−1
4. 4. Solve for 4−3y=2y+1
5. 5. Solve the equation: x2​ +1=3
6. 6. Solve the equation: 2z=3z−1
7. 7. Solve for 40−3y=12y
8. 8. Solve the equation: y2​ +1=3

Linear Equations: FAQs

What is a linear equation?

A linear equation is an algebraic expression where each term has a variable raised to the power of 1. It forms a straight line when graphed. Examples include 3x+2=8 and 2x−y=4.

How do you solve a linear equation in one variable?

To solve a linear equation in one variable, isolate the variable by performing inverse operations. For example, if the equation is 4x+5=17, subtract 5 from both sides to get 4x=12, then divide both sides by 4 to find x=3.

How can you check if your solution to a linear equation is correct?

To verify the solution, substitute the value back into the original equation. If both sides of the equation are equal, the solution is correct. For example, if the solution is x=3 for the equation 2x+4=10, substitute 3 in place of x: 2(3)+4=10, which confirms that the solution is correct.