Math Lesson 9.7.1 - Linear Equations Definition

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Welcome to our Math lesson on Linear Equations Definition, this is the first lesson of our suite of math lessons covering the topic of Systems of Linear Equations. Methods for Solving Them., you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

What are Linear Equations?

Linear equations are first - order equations with two variables. They are called "linear" because their graph is a straight line. Let's prove this statement.

Let's consider a first - order equation with two variables, for example 2x - y + 1 = 0. To solve this equation means to find all possible combinations of the variables x and y that make the left part of the equation zero, i.e. that give a true result. Obviously, there is an infinity of combinations that meet this condition. All these points lie on the graph. Below we are giving a few of them.

For x = 0, we have 2 ∙ 0 - y + 1 = 0, so y = 1. Hence, the number pair (0, 1) is a point of the graph.

For x = 1, we have 2 ∙ 1 - y + 1 = 0, so y = 3. Hence, the number pair (1, 3) is a point of the graph.

For x = 2, we have 2 ∙ 2 - y + 1 = 0, so y = 5. Hence, the number pair (2, 5) is a point of the graph.

For x = 3, we have 2 ∙ 3 - y + 1 = 0, so y = 7. Hence, the number pair (3, 7) is a point of the graph.

For x = 4, we have 2 ∙ 4 - y + 1 = 0, so y = 9. Hence, the number pair (4, 9) is a point of the graph, and so on.

Let's plot all the above points in a coordinates system.

It is clear that all these points are collinear. We can connect them with each other so that we obtain a straight line known as a linear graph, as shown below.

Obviously, we can extend the line beyond the points we found earlier. Moreover, it is worth highlighting the fact that not only integers are included in the graph, it contains all possible pairs of real numbers that make the equation true as well.

In fact, from geometry, it is known that we need to have only two points known to draw a line. Therefore, with just two pairs of coordinates, we can successfully plot a linear graph. It is preferred that these two points be the intercepts with the X - and Y - axes, i.e. the points A(0, y) (the Y - intercept) and B(x, 0) (the X - intercept), but it is up to you to choose the points you see as the most appropriate.

Example 1

Check whether points A(3, - 4), B( - 5, 0) and C( - 2, 5) lie on the graph of the linear equation x - 2y + 5 = 0.

Solution 1

All we have to do is to substitute the coordinates of each point in the equation and check whether they give a true result, i.e. the left part of the equation must be zero after substitutions. Thus, for point A (x = 3; y = - 4), we have

This means that point A is not a point on the graph of our equation.

As for point B (x = - 5; y = 0), we have

This means that point B is a point on the graph of our equation.

Last, for point C (x = - 2; y = 5), we have

This means that point C is not a point on the graph of our equation.

The three points A, B and C as well as the graph of the linear equation x - 2y + 5 = 0 are shown in the graph below.

More Systems of Linear Equations. Methods for Solving Them. Lessons and Learning Resources

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1. Definition Feedback. Helps other - Leave a rating for this definition (see below)
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3. Equations Video tutorial: Systems of Linear Equations. Methods for Solving Them.. Watch or listen to the Systems of Linear Equations. Methods for Solving Them. video tutorial, a useful way to help you revise when travelling to and from school/college
4. Equations Revision Notes: Systems of Linear Equations. Methods for Solving Them.. Print the notes so you can revise the key points covered in the math tutorial for Systems of Linear Equations. Methods for Solving Them.
5. Equations Practice Questions: Systems of Linear Equations. Methods for Solving Them.. Test and improve your knowledge of Systems of Linear Equations. Methods for Solving Them. with example questins and answers
6. Check your calculations for Equations questions with our excellent Equations calculators which contain full equations and calculations clearly displayed line by line. See the Equations Calculators by iCalculator™ below.
7. Continuing learning equations - read our next math tutorial: Relationship between Equations in Linear Systems. Systems of Equations with One Linear and One Quadratic Equation

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Equations Calculators by iCalculator™

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