Linear Equations Lesson - Definition, Formula, Graph, Examples

Lesson Overview

• What Is a Linear Equation?
• Linear equation and Non - Linear equations
• Linear Equation Formula
• Forms of Linear Equation
• How To Solve Linear Equations?
• Linear Equation Graph
• Linear Equation Examples

Linear equations are essential in math and science because they show how one thing changes in relation to another. They are used to solve real-world problems and are the foundation for more advanced math.

What Is a Linear Equation?

A linear equation is an algebraic equation that makes a straight line when graphed. The variables only have a highest power of 1.

Key Characteristics-

Variables with exponents of 1: For example, x and y, but not x² or y³.  

Represents a straight line: The line can have any direction (increasing, decreasing, horizontal, or vertical).

General Form:

A linear equation in two variables can often be written in the following form:

ax + by + c = 0

Where:

• x and y represent the variables.
• a and b are coefficients (numbers multiplied by the variables).  
• c is a constant. 
  

Example:

2x + 3y - 6 = 0

This equation, when graphed, will form a straight line. Linear equations are used to model relationships where the change in one variable is directly proportional to the change in another.

Linear equation and Non - Linear equations

The difference between linear and non-linear equations is that linear equations have a degree of 1 and form a straight line when graphed; non-linear equations have a degree of 2 or higher and form curves or other non-linear shapes when graphed.

Example	Linear or Non-linear
y = 2x + 5	Linear
y = x² + 3	Non-linear
3x - 4y = 12	Linear
y = √x	Non-linear
y = 4	Linear
x³ + y² = 10	Non-linear
y = 1/x	Non-linear
x = 7	Linear

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Linear Equation Formula

A linear equation formula is a mathematical expression that represents the relationship between variables in a linear equation. These formulas provide a structured way to define and work with straight lines on a graph. They allow us to analyze key characteristics of the line, such as its slope, intercepts, and the relationship between the variables.

Forms of Linear Equation

Linear equations can be expressed in various forms, each serving different purposes and providing unique insights into the line's properties.

Slope-Intercept Form

The slope-intercept form of a linear equation directly shows the slope and y-intercept of the line.

y = mx + b

• 'm' represents the slope of the line, which can be calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁) where (x₁, y₁) and (x₂, y₂) are any two points on the line.
• 'b' represents the y-intercept, which is the point where the line crosses the y-axis.
  

Example:

y = 2x + 3

In this equation:

• The slope (m) is 2, meaning the line goes up 2 units for every 1 unit it moves to the right.
• The y-intercept (b) is 3, meaning the line crosses the y-axis at the point (0, 3).
  
  

Point-Slope Form

The point-slope form is useful when you know the slope of a line and one point that lies on the line.

y - y₁ = m(x - x₁)

• 'm' is the slope of the line.
• (x₁, y₁) is a known point on the line.
  

Example:

A line has a slope of -1 and passes through the point (2, 4). Its equation in point-slope form is:

y - 4 = -1(x - 2)

This equation can be simplified to y = -x + 6, which is in slope-intercept form. It shows that the line has a slope of -1 and a y-intercept of 6.

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How To Solve Linear Equations?

Solving a linear equation means finding the value(s) for the variable(s) that make the equation true. To do this, there is a need to isolate the variable and get on one side of the equation.

Linear Equations in One Variable

These equations have a single unknown variable.

• Example: 2x + 5 = 11
  
• Subtract 5 from both sides: 2x = 6
• Divide both sides by 2: x = 3
  

Linear Equations in Two Variables

These equations have two unknowns. A solution is a pair of values (x, y) that satisfies the equation.

• Example: y = 3x - 2 and x + y = 4
  
• Since we know y = 3x - 2, substitute that into the second equation: x + (3x - 2) = 4
• Simplify and solve for x = 4x - 2 = 4 => 4x = 6 => x = 3/2
• Substitute x back into either original equation to find y = (3/2) + y = 4 => y = 5/2
  

Linear Equations in Three Variables

These equations have three unknowns. A solution is a set of values (x, y, z) that satisfies the equation.

• Example:
  
  • x + y + z = 6
  • 2x - y + z = 3
  • x + 2y - z = 2
    
• This requires multiple steps using elimination and substitution (or matrices):
  
• To eliminate 'z' add the first and third equations = 2x + 3y = 8. 
• Add the first and second equations: 3x = 9.
• Solve for 'x' from 3x = 9, we get x = 3.
• Substitute to find 'y', substitute x = 3 into 2x + 3y = 8 to get 6 + 3y = 8 => 3y = 2 => y = 2/3
• Substitute to find 'z', substitute x = 3 and y = 2/3 into x + y + z = 6 to get 3 + 2/3 + z = 6 => z = 7/3

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Linear Equation Graph

The graph of a linear equation in one variable, x, forms a vertical line parallel to the y-axis. On the other hand, the graph of a linear equation in two variables, x and y, forms a straight line. Let's graph the following linear equation:

Example: Graph the linear equation x - 2y = 2

Step 1: The given equation is:

x - 2y = 2

Step 2: Rearrange the equation to the form y = mx + b.

Solve for y:

-2y = 2 - x
y = (x / 2) - 1

Now the equation is in the form y = (x / 2) - 1.

Step 3: Choose different values for x and calculate the corresponding y values.

Step 4: Calculate the coordinates for different values of x:

• When x = 0:
  y = (0 / 2) - 1 = -1
  So, the point is (0, -1).
  
• When x = 2:
  y = (2 / 2) - 1 = 0
  So, the point is (2, 0).
  
• When x = 4:
  y = (4 / 2) - 1 = 1
  So, the point is (4, 1).
  
• When x = -2:
  y = (-2 / 2) - 1 = -2
  So, the point is (-2, -2).

Step 5: List the coordinates:

X	Calculation	Y
0	y = (0 / 2) - 1	-1
2	y = (2 / 2) - 1	0
4	y = (4 / 2) - 1	1
-2	y = (-2 / 2) - 1	-2

Step 6: Plot the points (0, -1), (2, 0), (4, 1), and (-2, -2) on a graph.

• Connect the points to form a straight line.

This line represents the graph of the equation x - 2y = 2.

Linear Equation Examples

Example 1: 2x + 5 = 11

Solution

• Subtract 5 from both sides: 2x = 6
• Divide both sides by 2: x = 3
  

Example 2: 4y - 7 = 9

Solution

• Add 7 to both sides: 4y = 16
• Divide both sides by 4: y = 4
  

Example 3: 3(z + 2) = 15

Solution

• Distribute the 3: 3z + 6 = 15
• Subtract 6 from both sides: 3z = 9
• Divide both sides by 3: z = 3
  

Example 4: (a/5) + 1 = 3

Solution

• Subtract 1 from both sides: a/5 = 2
• Multiply both sides by 5: a = 10
  

Example 5: 2(b - 4) = b + 6

Solution

• Distribute the 2: 2b - 8 = b + 6
• Subtract b from both sides: b - 8 = 6
• Add 8 to both sides: b = 14