Dependent System of Linear Equations Examples

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  • 0:03 Defining System of Equations
  • 2:07 Identifying Dependent Systems
  • 3:53 Algebra with Dependent Systems
  • 5:14 Lesson Summary
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Lesson Transcript
Instructor: Mia Primas

Mia has taught math and science and has a Master's Degree in Secondary Teaching.

What do you think of when you hear the word 'dependent?' You probably think of dependent children or someone that relies on another person. Well, equations can be dependent, too. In this lesson, you'll learn how to identify dependent systems of linear equations.

Defining System of Equations

First, let's go over the definition of 'system of equations'. A system of equations is two or more equations that are solved simultaneously. To solve the system, you must find solutions for each variable. In a system of linear equations, the equations are all linear, meaning that they form a straight line when graphed. The graphs of the equations may be parallel, intersect at one point, or form the same line. If the equations form the same line, we refer to it as a dependent system of linear equations.


The two equations in this system appear to be completely different. But when they are graphed, we see that they produce the same line. The solution to a system of equations lies at the point of intersection.

So how can we identify the solution to a dependent system? Since the lines are the same, we can think of them as intersecting at every point, meaning that every solution for x and y that satisfies one equation will also satisfy the other.

For example, we can substitute the coordinates (1, 3) in the first equation, giving us 3 = 1 + 2. Since the equation is true, we know that (1, 3) is a solution for this equation. If we substitute the same values in the second equation, we get -3(1) + 3(3) = 6. After simplifying it to -3 + 9 = 6, we see that the coordinates also satisfy the second equation.

We could continue to test any point that satisfies one equation and see that it also satisfies the other. This creates an infinite number of solutions for the system. In fact, all dependent systems have an infinite number of solutions.

In this example, we were able to use the graph to determine whether the system was dependent or not. Let's see how we can identify dependency by examining the equations.

Identifying Dependent Systems

There are several ways that we can determine whether a system of equations is dependent without graphing the lines. We could use trial and error, testing several solutions for one equation to see if they satisfy the other, but that would be time consuming. Another strategy involves finding the slopes and intercepts of the equations.

If the slopes and intercepts are the same, then they must be the same line. The first equation in our previous example was in the form y = mx + b, also called slope-intercept form. The coefficient of x is the slope, or m, and the constant term is the y-intercept, or b. This allows us to quickly identify the slope as 1 and the y-intercept as 2.

We can rewrite the second equation in slope-intercept form by adding 3x and dividing by 3 on both sides, resulting in y = x + 2. Without going any further, we see that the slope-intercept form of the second equation is exactly the same as the first equation. But just to be sure, we can compare their slopes and y-intercepts. Both equations have a slope of 1 and a y-intercept of 2. This tells us that they are the same line and that the system of equations is dependent.

We can also tell that a system of equations is dependent if one equation is a multiple of the other.


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