Introduction to Linear Equations

A linear equation is an equation in which the highest power (degree) of the variable is 1.

• It represents a straight line when graphed.
• It can have one, two, or more variables, but they do not include exponents or higher powers.
• The linear equation in one variable represents a straight line parallel to either axis.

General Forms:

• One variable: ax + b = 0 (a ≠ 0)
• Two variables: Ax + By + C = 0

Here, a, b, c are constants and x, y are variables.

Examples of Linear Equations

1. One Variable: A linear equation involving one variable has the form ax + b = 0. It can be solved using a single equation. Example: x + 4 = 6

2. Two Variables: A linear equation involving two variables has the form ax + by = c. A single equation represents a line and has infinitely many solutions; two such equations are required to get a unique solution. Example: x + y = 6

3. Three Variables: A linear equation involving three variables has the form ax + by + cz = d. A single equation represents a plane and has infinitely many solutions; three independent equations are required to get a unique solution.
Example : x + y + z = 6

Representations or Different Forms

There are various ways to represent the linear equations, such as

• Standard Form: ax + by = c. Example, 2x + 3y = 6
• Slope-Intercept Form: y = mx + b. Example, y = 2x + 3
• Point-Slope Form: y - y₁ = m(x - x₁). Example, y - 4 = 3(x - 2)
• Intercept Form: x/a + y/b = 1, Example, x/2 + y/3 = 1

Graphing Linear Equations

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.

Solved Examples

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,

• x = -1
• y = -6

Example 3: 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 x from 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

• x = 3
• y = 0

Practice Questions

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.