Linear Dependence & Independence: Definition & Examples

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Linear dependence and independence are based on whether or not there is more than one solution to a system of equations. In this lesson, we'll look at how you can determine whether or not a system is independent and work through some examples.

What are Linear Dependence and Independence?

''Mr. Smith, I just can't get an answer for this system of equations. No matter what I do, both variables vanish, and I end up with the same number on both sides of the equation.''

Ever run across this situation with a system of equations? You use the appropriate combination of substitution and/or elimination methods, yet the solution seems elusive, disappearing completely while you're working. You may be in a situation where the solution for the equation is ''dependent'', which means that the correct y depends on what x value you're using.

Dependence in systems of linear equations means that two of the equations refer to the same line. There is an infinite number of solutions that will satisfy the conditions of the equations! To know which solution you want, you have to feed in an x value. This makes the y value dependent on the x value.

Independence in systems of linear equations means that the two equations only meet at one point. There is only one point in the entire universe that will solve both equations at the same time. It's the intersection between the two lines.

Testing Equations for Dependence and Independence

  1. If the slopes are different, the system is independent
  2. If the slopes are the same, then the system is either dependent (same line) or inconsistent (parallel lines)

There are multiple ways to find out if a system of linear equations is dependent or independent, and it depends on what you like to do. For example, if you're one of those ''plug and chug'' types who loves to work algebraic equations, you can find out quickly just by solving both equations for y (or whatever output variable you're using). This will put the equations in slope-intercept form, which means all you have to do is compare the coefficient in front of the x term, the slope. If the two slopes are different, then the two lines will collide somewhere, and you have an independent system. If the slopes are the same, then you either have a dependent system (the two equations are the same line) or an inconsistent system with no solution (the two lines are parallel and will never meet).

If you happen to be the artistic type who loves to graph lines, then you can use the intercepts, T-table, slope-intercept, or whatever approach you like, and graph the two lines. If the two lines end up being the same line, then you have a dependent situation. Otherwise, check to see if they're parallel. If they're not parallel, then you have an independent solution; even if they don't cross on your graph, they will certainly cross somewhere!


Given the following system of linear equations, you can take more than one approach to determine whether or not the solution is independent:

4x + 2y = 6

y = -x - 2

We can solve for y in the first equation and then compare slopes.

4x + 2y = 6

2y = -4x + 6 (Subtract 4x from both sides)

y = -2x + 3 (Divide both sides by 2)

The slope is the coefficient (the number being multiplied by the variable) in front of the x. In the first equation, the slope turns out to be -2, while in the second equation the slope is -1. Since the two slopes are different, these two lines will certainly meet somewhere, so this system has an independent solution.

If you choose to graph these two lines, you can see that they are definitely independent, having only one place where they cross.

graphic image

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