We often use linear equations in our daily lives, whether we’re aware of it or not. For instance, suppose you are planning a birthday party for a friend. It costs £375 to hire a venue, and it will cost a further £14.50 per person for food.
To formulate how much the total cost of the party will be, you will use a linear equation: y = 14.50x + 375. The total cost would be represented by y, and the total number of party guests would be represented by x.
If there’s any situation where there’s an unknown variable, you can almost always formulate it using a linear equation. That’s why there’s such a heavy focus on it in school curriculums; it has real-world applications.
To help you understand what it is, we’ve created this article to explain all things linear equation-related such as the different forms it comes in and how to solve them. Let’s dive in.
A linear equation is an equation where each term has the highest degree of 1, and when shown in a graph, it is represented as a straight line. When solving linear equations, if you graph linear equations and plot the solutions in a coordinate system, it will make a straight line. The straight line is what makes the equation ‘linear’, hence why they are called linear equations.
There are different forms of linear equations, but the most common one is the slope intercept form, which follows the format y = mx + b. But before we explain what it is, let’s take a look at all the different forms of linear equations.
What are the different forms of linear equations?
There are four main forms of linear equations – standard form, slope intercept form, point slope form, and a linear function. Let’s take a look at these in more detail.
The standard form – also known as the general form – of a linear equation in one variable is as follows:
- ax + b = 0
In this example, x is the only variable, and a and b are real numbers. Also, a cannot equal zero.
The standard form of linear equation in two variables is as follows:
- ax + by + c = 0
In this example, x and y are the two variables, and a, b, and c are real numbers. Also, a and b cannot equal zero.
Alongside linear equations in two variables, The standard form of linear equations in three variables is as follows:
- ax + by + cz + d = 0
In this example, x, y, and z are the three variables, and a, b, c, and d are real numbers. Also, a, b, and c cannot equal zero.
Slope intercept form
The slope intercept form is perhaps the most common type of linear equation. It’s what students will come across most when solving linear equations. The slope intercept form is as follows:
- y = mx + b
In this example, m is the slope of the line, also known as the gradient. It describes if the line increases or decreases as it moves along the x-axis. b is the y-intercept, meaning that’s when it crosses the y-axis. Lastly, x and y are simply coordinates of their respective axis.
To help you visualize the slope intercept form, let’s take an example of y = b. In this example, there is no slope of the line since x = 0, which means that we have a straight line. The horizontal line will intercept the y-axis at whatever value b is.
Conversely, if we have mx + b = 0, this means that we have a vertical line that is parallel to the y-axis since y = 0.
Point slope form
We also have the point slop form. The point slope form is as follows:
- y – y1 = m(x – x1 )
In this example, both y1 and x1 are coordinates of the point. Sometimes you may see the point slop form expanded as a function of y, which looks like:
- y = mx + y1 – mx1
Lastly, we have linear equations that are written as a linear function. Linear functions typically start with f(x) and look something like this:
- f(x) = ax + b
In this example, y has been substituted for f(x). As you can see, it’s not too dissimilar from the standard form of linear functions.
Linear equations examples
Examples of linear equations are as follows:
- y = 6x - 5
- 5y = 2x + 3
- 4x + 7y = 63
- 2x + 3y - 8z + 3 = 85
- y + 9x - 47 = 0
- 76x = 38
These all classify as linear equations because the highest degree of each variable is 1 – i.e., no terms are squared, cubed, or square rooted.
Examples of non-linear equations are as follows:
- y = x2 + 3
- y3 + 2x = 7
- √y - x + 5z = 6
These are all non-linear equations because the highest degree of some variables does not equal 1. We have an x2, y3, and √y, all of which do not satisfy the linear equation definition.
Now we know what linear equations are, the different forms they come in, and examples of what they look like. But how do we solve them?
Simply isolate the x and solve. If you have two unknown variables, such as x and y, you will have to solve for one and then solve for the other. Let’s take a look at some examples.
Example 1: solve 4x - 3 = 21
This example shows the linear equation in its slope intercept form. To solve this, you must balance both sides of the equation.
The first step involves isolating the x. To do this, you have to add 3 to both sides of the equation. This cancels out the -3 on the left side of the equation to give you the following equation:
- 4x = 24
From here, you want to divide both sides of the equation by 4, as this will isolate x on its own. This gives us the equation x = 6. Since the x has been isolated, we have reached our final answer:
- x = 6
Example 2: solve y = 6x - 5
We’ve seen an example of how to solve linear equations by isolating x, but you can also solve it by classing the solutions as coordinates when y = 0. The coordinates would be as follows:
(-b/a , 0)
The -b/a represents the x coordinate, and the 0 represents the y coordinate.
For instance, suppose we want to solve the y = 6x - 5. If y = 0, we can substitute 0 into the equation to give us the following:
- 0 = 6x - 5
Then, we would add 5 to both sides and finally divide both sides by 6 to isolate x. This means that x would equal:
- x = 5/6
If you notice, the 5/6 represents -b/a. Therefore, when writing our answer as a coordinate, it would be as follows:
- (⅚ , 0)
Linear equations are an equation term that has a highest degree of 1, and when displayed on a graph, is a straight line – which is what makes the equations 'linear'. There are various forms of linear equations, but the most common types are the standard form and slope intercept form.
Solving for linear equations involves isolating the variable – usually x – and simplifying it until you have found its value. You can also represent the solutions as a coordinate when y = 0.