Since the slopes are different, the lines will intersect at exactly one point. The two lines y = -3 (blue) and x = 2 (red) have different slopes, so they intersect at one point, and there is one solution to the linear system. Now we can substitute x = -1 into the first equation to get: We’ll substitute the y from the first equation into the y in the second equation: Example 2: Using Substitution To Show A Linear System Has One Solution The two lines 2x + 4y = 3 (blue) and -6x – 8y = 11 (red) have different slopes, so they intersect at one point, and there is one solution to the linear system. The graph below confirms that the lines have different slopes and intersect exactly once. Since there is only one ordered pair that makes both equations true, there is only one solution to this system of linear equations. We can test this by substituting x = -8.5, y = 5 into both of the original equations to make sure we get the right values. Substituting y = 5 into the first equation gives us: Now we add this modified equation to the second one: We begin by multiplying the first equation by 3 to get: Let’s say we want to solve the following system of linear equations: Example 1: Using Elimination To Show A Linear System Has One Solution Let’s take a look at some examples to see how this can happen. For example, after we solve, we will get something like x = 2 and y = 5. When we solve a linear system with one solution, we will get a result that gives us a single value for x and a single value for y. Solving A Linear System With One Solution We’ll look at some examples of each case, starting with solving the system.
MULTIVARIABLE EQUATION SYSTEMS OF EQUATIONS HOW TO
If the two equations have different slopes, then the two lines are different and are not parallel, so there is only one solution ( you can get a refresher on how to tell when two lines are parallel in my article here).
Solve the system – if you solve the system and get a single equation (such as x = 2 and y = 5), then there is one solution.There are a few ways to tell when a linear system in two variables has one solution: When Does A Linear System Have One Solution? (System Of Linear Equations In 2 Variables) We’ll start with linear equations in 2 variables with one solution. The image below summarizes the 3 possible cases for the solutions for a system of 2 linear equations in 2 variables.Ī system of 2 linear equations in 2 variables has one solution when the two lines have different slopes.Ī system of equations in 2, 3, or more variables can have one solution. This means that there is one point that can satisfy all of the equations at the same time. Systems Of Linear Equations With One SolutionĪ system of linear equations can have one solution if there is a single point that makes every equation in the system true. We’ll also look at some examples of linear systems with one solution in 2 variables and in 3 variables. In this article, we’ll talk about how you can tell that a system of linear equations has one solution. Of course, a system of three equations in three variables has one solution if there is exactly one point where all three planes intersect.
From an algebra standpoint, this means that we get a single value when solving the system. Visually, the lines intersect exactly once on a graph, since they have different slopes. So, when does a system of linear equations have one solution? A system of two linear equations in two variables has one solution when the two lines have different slopes. However, it is also possible that a linear system will have exactly one solution. When working with systems of linear equations, we often see infinitely many solutions or one at all.
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