A linear equation is a fundamental concept in algebra that represents a straight line in coordinate geometry. It is a first-order equation, meaning the highest power of the variable is 1. Linear equations can have one, two, or more variables, but they do not include exponents or higher powers.
The standard form of the linear equation is
Ax + By + C = 0
Where A, B, and C are constants.
Linear EquationsExamples of Linear Equations
Some examples of linear equations are in the table below:
| Linear Equations with One Variable | Linear Equations with Two Variables | Linear Equations with Three Variables |
|---|
| Linear equations, i.e. the equation with power one and with one variable are called the linear equation with one variable. | Linear equations, i.e. the equation with power one and with two variables are called the linear equation with two variables. | Linear equations, i.e. the equation with power one and with three variables are called the linear equation with three variables. |
| To find the unique solution to these equations only one equation is required. | To find the unique solution to these equations two equations are required. | To find the unique solution to these equations three equations are required. |
| Example: x + 4 = 6 | Example: x + y = 6 | Example: x + y + z = 6 |
The formula used to represent the linear equations is called the linear equation formula. There are various ways to represent the linear equations such as,
Equation Form | Equation | Example |
|---|
Standard Form | ax + by = c | 2x + 3y = 6 |
|---|
Slope-Intercept Form | y = mx + b | y = 2x + 3 |
|---|
Point-Slope Form | y - y₁ = m(x - x₁) | y - 4 = 3(x - 2) |
|---|
Intercept Form | x/a + y/b = 1 | x/2 + y/3 = 1 |
|---|
Standard Form of Linear Equation
Standard form of the linear equation in two variables is given in the image below:
Linear Equation with coefficientsWhere A and B are never zero.
Some examples of the linear equation in two variables in standard form are,
Slope-Intercept form of the linear equation in two variables is,
y = mx + c
where
- m is the slope of the line, and
- c is the intercept on the y-axis.
Some examples of the linear equation in slope-intercept in two variables are,
Point-Slope form of the linear equation in two variables is,
y - y1 = m(x - x1)
Where,
- m is the slope of the line, and
- (x1, y1) is the given point.
Some examples of the linear equation in slope-intercept in two variables are,
- y - 11 = 3(x - 6)
- y + 5 = 4(x + 11)
Intercept form of the linear equation in two variables is,
x/a + y/b = 1
Where a and b are the x and y intercepts cut by the graph of the line.
Some examples of the linear equation in intercept form for two variables are,
- x/2 + y/3 = 1
- x/3 - y/2 = 1
- y/7 - x/4 = 1
How to Solve Linear Equations?
The solution to the linear equation is mostly studied under two heading
- Solution of Linear Equations in One Variable
- Solution of Linear Equations in Two Variable
Let's learn about both in detail.
Solving Linear Equations in One Variable
To solve the linear equation in one variable we first isolate the linear equation where all the variables are on the LHS and the constants are on the RHS and each side is individually simplified and then solved to get the required solution. This can be understood with the help of the example discussed below.
Example: Solve the equation, 4x - 6 = 2 + 2x
Solution:
Given,
4x - 6 = 2 + 2x
Taking all the variables on the LHS and the constants on the RHS
4x - 2x = 2 + 6
⇒ 2x = 8
⇒ x = 4
This is the required solution of the given equation.
Solving Linear Equations in Two Variables
To solve the linear equation in two variables we need two linear equations that are solved to get the solution of the linear equation. There are various methods to solve the linear equations in two variables such as
- Graphical Method
- Elimination Method
- Substitution Methods
- Cross Multiplication Method
The solution of a linear equation in two variables gives coordinates of a point that lies on the line defined by that linear Equation.
Example: Solve the equation, x + y = 12 and y = x - 2.
Solution:
Given Equation,
• x + y = 12...(i)
• y = x - 2...(ii)
putting the value of y from eq (ii) in eq (i) we get
x + (x - 2) = 12
⇒ x + x - 2 = 12
⇒ 2x = 12 + 2
⇒ 2x = 14
⇒ x = 7
Putting the value of x in eq (i)
7 + y = 12
⇒ y = 12 - 7 = 5
Thus,
• x = 7
• y = 5
Linear Equation Graph
The graph of the linear equation generally represents a straight line. The linear equation in one variable represents a straight line parallel to either axis, this can be understood as, x + 7 = 0
This linear equation in one variable represents a straight line passing through the point (-7, 0) and parallel to the y-axis. Similarly linear equations in two variables also represent a straight line and its graph can be plotted by following the steps discussed below,
Example: Plot the graph for a linear equation in two variables, x + y - 6 = 0
Use the following steps to plot the graphs
Step 1: Arrange the given equation of the line in the standard form as, x + y = 6
Step 2: Now change the equation in the intercept form by dividing 6 on both sides to make the RHS 1.
x/6 + y/6 = 1
Step 3: The denominator of x and y represents the intercept on the x and y axis respectively. The intercept on the x-axis is 6 and the intercept on the y-axis is 6.
Step 4: Find the point on the x-axis and the y-axis, i.e. the point on the x-axis is (6, 0) and the point on the y-axis is (0, 6). Join these points to get the line.
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Solved Examples of Linear Equations
Example 1: Solve 2x = 3(x + 4)
Solution:
Given equation,
2x = 3(x + 4)
⇒ 2x = 3x + 3(4)
⇒ 2x = 3x + 12
⇒ 2x - 3x = 12
⇒ -x = 12
⇒ x = -12
Example 2: Solve 2x – y = 4 and x + y = 5
Solution:
Given Equation,
- 2x – y = 4...(i)
- x + y = 5...(ii)
From eq. (ii) y = 5 - x
Putting the value of y from eq (ii) in eq (i) we get
2x + (5 - x) = 4
⇒ 2x + 5 - x = 4
⇒ 2x - x = 4 - 5
⇒ x = -1
Putting value of x in eq (i)
2(-1) - y = 4
⇒ -2 - y = 4
⇒ y = -2 - 4
⇒ y = -6
Thus,
Example 3: Solve x - 7 = 2(x - 3)
Solution:
Given equation,
x - 7 = 2(x - 3)
⇒ x - 7 = 2x - 6
⇒ -7 + 6 = 2x - x
⇒ x = -1
Example 4: Solve 2x + 3y = 6 and x - y = 3.
Solution:
Given Equation,
- 2x + 3y = 6 . . .(i)
- x - y = 3 . . .(ii)
take equation (ii),
x = y + 3 . . .(iii)
putting the value of xfrom eq (iii) in eq (i) we get
2(y + 3) + 3y = 6
⇒ 2y + 6 + 3y = 6
⇒ 5y = 6 - 6
⇒ 5y = 0
⇒ y = 0
Putting the value of y in eq (iii)
x = y + 3 = 0 + 3 = 3
Practice Questions on Linear Equations
Question 1: Solve the linear equation: 3x + 7 = 19.
Question 2: Find the value of y in the equation: 2y - 5 = 3y + 8.
Question 3: If the sum of two numbers is 15, and one of the numbers is 7, form a linear equation and find the other number.
Question 4: Solve for x in the equation: \frac{2x}{3} + 4 = 10
Question 5: A mobile phone plan costs a fixed monthly charge of $30 plus $0.10 per minute of usage. If a user has a bill of $50, how many minutes did they use? Form and solve the linear equation.
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