Linear Equations: Definition, Formula, Graph, Examples

Linear equations are fundamental in mathematics and are one of the most commonly used forms of mathematical expression. They are equations of the first order, meaning that they contain no exponents or roots. Typically, a linear equation involves one or two variables and presents a direct relationship between these variables.

What is a Linear Equation?

In simple terms, a linear equation is an equation that describes a straight line on a graph. The word ‘linear’ comes from the Latin word ‘linea’, which means ‘line’. A linear equation can be expressed in various forms, but the most common form is the slope-intercept form, represented as y = mx + c.

In this equation, m is the slope of the line, c is the y-intercept, and x and y are the variables or coordinates on the x-axis and y-axis respectively.

Describing Linear Relationships

A linear relationship is a relationship between two variables in which the rate of change of one variable is constant in relation to the other. In the context of linear equations, the variable y is often dependent on x. This means that the value of y changes according to the value of x.

For example, if we have a linear equation y = 2x + 1, for every unit increase in x, y will increase by twice that amount. This direct proportionality between x and y is a defining characteristic of linear relationships.

Linear Equation Formula

The formula for a linear equation can be presented in several forms. Each form serves a specific purpose and can provide different information about the line.

Standard Form: Ax + By = C, where A, B, and C are real numbers, and x and y are variables. This form is useful for finding the x and y intercepts of the line.
Slope-Intercept Form: y = mx + c, where m is the slope and c is the y-intercept. This form is useful for understanding the steepness and direction of the line.
Point-Slope Form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line. This form is useful for finding the equation of a line when given a single point and the slope.

How To Find Linear Equations?

The process of finding a linear equation depends on the information provided. If the slope of the line and the y-intercept are known, you can directly substitute these values into the slope-intercept form: y = mx + c.

If, however, you are given two points on the line, (x₁, y₁) and (x₂, y₂), you can use these points to calculate the slope (m) using the formula (y₂ - y₁) / (x₂ - x₁). Once the slope is known, it can be substituted into the point-slope form to find the equation of the line.

For those intrigued by the dynamics of linear equations, our dedicated slope page offers a valuable resource, delving into the fundamental concept of slope and its pivotal role in understanding and interpreting the graphical representation of linear equations.

Linear Equation Graph

The graph of a linear equation is a straight line. Each point on this line corresponds to a solution of the equation. The line extends infinitely in both directions because a linear equation in two variables has an infinite number of solutions.

The slope of the line determines its steepness and direction. A positive slope indicates an upward trend, while a negative slope indicates a downward trend. A slope of zero implies a horizontal line, while an undefined slope implies a vertical line.

What is Slope?

The slope of a line is a measure of the steepness or incline of the line. It is a fundamental concept in the graphing of linear equations. The slope is usually represented by the letter m and is calculated by the formula m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are any two points on the line.

Calculating Slope

To calculate the slope of a line:

Choose any two points on the line. Let’s call these points A and B.
Subtract the y-coordinate of point A from the y-coordinate of point B. This gives you the change in y, represented as ∆y or (y₂ – y₁).
Similarly, subtract the x-coordinate of point A from the x-coordinate of point B. This gives you the change in x, represented as ∆x or (x₂ – x₁).
The slope of the line is then given by m = ∆y / ∆x.

Positive Slope

A line has a positive slope if it slopes upwards from left to right. This means that as x increases, y also increases. The steeper the line, the greater the slope. For instance, a line with a slope of 3 is steeper than a line with a slope of 2.

Negative Slope

A line has a negative slope if it slopes downwards from left to right. This means that as x increases, y decreases. Again, the steepness of the line is determined by the value of the slope. A line with a slope of -3 is steeper than a line with a slope of -2.

Zero Slope

A line has a zero slope if it is horizontal. This means that y does not change as x changes. In other words, no matter what the value of x is, y remains constant.

Undefined Slope

A line has an undefined slope if it is vertical. This means that x does not change as y changes. In other words, no matter what the value of y is, x remains constant.

Lines With the Same Slope

Two lines that have the same slope are either parallel lines or the same line. In other words, if two lines have the same slope, they will never intersect and are therefore parallel. If two lines have the same slope and the same y-intercept, they are the same line.

Solving Two-Step Linear Equations with Rational Numbers

A two-step linear equation is an equation that requires two operations to be solved. These operations can be addition, subtraction, multiplication, or division. The solution to a two-step linear equation involves finding the value of the variable.

Forms of Linear Equation

Linear equations can be represented in various forms. Each form provides a different perspective and can be used in different situations.

Standard Form

The standard form of a linear equation is Ax + By = C, where A, B, and C are real numbers and x and y are variables. This form is useful for finding the x and y intercepts of the line.

Slope-Intercept Form

The slope-intercept form is y = mx + c, where m is the slope and c is the y-intercept. This form is useful for understanding the steepness and direction of the line.

Point Slope Form

The point-slope form is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line. This form is useful when you know the slope of the line and a point on the line.

How to Solve Linear Equations

The process of solving linear equations involves finding the values of the variables that make the equation true. There are several methods to solve linear equations, including substitution, elimination, and matrix methods.

Solution of Linear Equation in One variable

A linear equation in one variable is an equation with only one variable. It can be solved by isolating the variable on one side of the equation.

For example, to solve the equation 2x + 3 = 7, you would first subtract 3 from both sides to get 2x = 4, then divide both sides by 2 to get x = 2.

Solution of Linear Equation in Two variables

A linear equation in two variables is an equation with two variables. It can be solved by using methods like substitution, elimination, or graphing.

For example, to solve the system of equations x + y = 5 and x - y = 1, you could add the two equations together to get 2x = 6, then divide by 2 to get x = 3. Substituting x = 3 into the first equation gives 3 + y = 5, which simplifies to y = 2.

Solution of Linear Equations in Three Variables

A linear equation in three variables is an equation with three variables. It can be solved by using methods like substitution, elimination, or matrix methods.

For example, to solve the system of equations x + y + z = 6, 2x + 3y - z = 14, and 3x - y + 2z = 13, you could use the substitution or elimination method to find the values of x, y, and z.

Solving Linear Equations with Examples

Let’s take a look at a step-by-step example of how to solve a linear equation.

Example: Solve the equation 3x - 2 = 4.

Step 1: Add 2 to both sides of the equation to isolate the term with the variable. This gives 3x = 6.
Step 2: Divide both sides by 3 to solve for x. This gives x = 2.

How Do You Identify a Linear Equation?

A linear equation can be identified by the following characteristics:

The equation only involves two variables.
The variables are only to the first power.
There are no products of variables (like xy).
There are no variables in the denominator of a fraction.
There are no square roots, cube roots, or higher roots involving the variables.

What Is the Most Basic Linear Equation?

The most basic linear equation is y = x. This equation represents a straight line that passes through the origin (0, 0) and has a slope of 1. This means that for every unit increase in x, there is a corresponding unit increase in y.

Can Linear Equations Have Fractions?

Yes, linear equations can include fractions. However, the variable should not appear in the denominator of the fraction. For example, 2/3x + 1/2 = 1 is a valid linear equation, but 1/(2x) + 1 = 1 is not a linear equation because x is in the denominator of the fraction.

How Do You Convert a Linear Equation to Standard Form?

To convert a linear equation to standard form, you should rearrange the equation so that all terms involving the variables are on one side and the constant is on the other side. For example, the equation y = 2x + 3 in slope-intercept form can be converted to standard form as 2x - y = -3.

How Are Quadratic Equations Different from Linear Equations?

Quadratic equations and linear equations are both types of polynomial equations, but they differ in several ways. The most significant difference is that a quadratic equation has a term with a variable raised to the power of 2, while a linear equation only involves variables to the first power. Also, the graph of a linear equation is a straight line, while the graph of a quadratic equation is a parabola, which is a curved line.

For enthusiasts exploring the world of linear equations, our quadratic equations page serves as an engaging companion, providing a seamless transition into the realm of quadratic expressions and equations, fostering a holistic understanding of algebraic principles.

How to Graph Linear Equations?

Graphing a linear equation involves plotting points on a graph that satisfy the equation and then connecting these points to form a line. Here are the steps for graphing a linear equation:

Step 1: Choose values for x and substitute these values into the equation to find the corresponding values for y.
Step 2: Plot the points (x, y) on a graph.
Step 3: Draw a straight line that passes through these points.

The line represents all the solutions to the equation, and any point on the line will satisfy the equation.

By understanding the concept of linear equations, one can solve a wide range of problems in mathematics and physics. With practice, you can become proficient in identifying, solving, and graphing linear equations.