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If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Video transcriptWe're asked to use the drop-down to form a linear equation with infinitely many solutions. So an equation with infinitely many solutions essentially has the same thing on both sides, no matter what x you pick. So first, my brain just wants to simplify this left-hand side a little bit and then think about how I can engineer the right-hand side so it's going to be the same as the left no matter what x I pick. So right over here, if I distribute the 4 over x minus 2, I get 4x minus 8. And then I'm adding x to that. And that's, of course, going to be equal to 5x plus blank. And I get to pick what my blank is. And so 4x plus x is 5x. And of course, we still have our minus 8. And that's going to be equal to 5x plus blank. So what could we make that blank so this is true for any x we pick? Well, over here we have 5 times an x minus 8. Well, if we make this a minus 8, or if we subtract 8 here, or if we make this a negative 8, this is going to be true for any x. So if we make this a negative 8, this is going to be true for any x you pick. You give me any x, you multiply it by 5 and subtract 8, that's, of course, going to be that same x multiplied by 5 and subtracting 8. And if you were to try to somehow solve this equation, subtract 5x from both sides, you would get negative 8 is equal to negative 8, which is absolutely true for absolutely any x that you pick. So let's go-- let me actually fill this in on the exercise. So I want to make 5-- it's going to be 5x plus negative 8. When working with systems of linear equations, we often see a single solution or no solution at all. However, it is also possible that a linear system will have infinitely many solutions. So, when does a system of linear equations have infinite solutions? A system of two linear equations in two variables has infinite solutions if the two lines are the same. From an algebra standpoint, we get an equation that is always true if we solve the system. Visually, the lines have the same slope and same y-intercept (they intersect at every point on the line). Of course, a system of three equations in three variables has infinite solutions if the planes intersect in an entire line (or an entire plane if all 3 equations are equivalent). In this article, we’ll talk about how you can tell that a system of linear equations has infinite solutions. We’ll also look at some examples of linear systems with infinite solutions in 2 variables and in 3 variables. Let’s begin. A system of linear equations can have infinite solutions if the equations are equivalent. This means that one of the equations is a multiple of the other. It also means that every point on the line satisfies all of the equations at the same time. The image below summarizes the 3 possible cases for the solutions for a system of 2 linear equations in 2 variables. A system of equations in 2, 3, or more variables can have infinite solutions. We’ll start with linear equations in 2 variables with infinite solution. When Does A Linear System Have Infinite Solutions? (System Of Linear Equations In 2 Variables)There are a few ways to tell when a linear system in two variables has infinite solutions:
We’ll look at some examples of each case, starting with solving the system. Solving A Linear System With Infinite SolutionsWhen we attempt to solve a linear system with infinite solutions, we will get an equation that is always true as a result. For example, after we simplify and combine like terms, we will get something like 1 = 1 or 5 = 5. Let’s take a look at some examples to see how this can happen. Example 1: Using Elimination To Show A Linear System Has Infinite SolutionsLet’s say we want to solve the following system of linear equations:
We will use elimination to solve. Let’s try to eliminate the “x” variable. We begin by multiplying the first equation by 3 to get:
Now we add this modified equation to the second one: 6x + 12y = 9 + -6x – 12y = -9 ___________ 0x + y = 0 This implies 0 = 0, which is always true – regardless of the values of x or y we choose. This means that both equations represent the same line. It also means that every point on that line is a solution to this linear system. So, the system has infinite solutions. The graph below shows the line resulting from both of the equations in this system. Example 2: Using Substitution To Show A Linear System Has Infinite SolutionsLet’s say we want to solve the following system of linear equations:
We will use substitution to solve. We’ll substitute the y from the first equation into the y in the second equation:
This implies 0 = 0, which is always true – regardless of the values of x or y we choose. This means that both equations represent the same line. It also means that every point on that line is a solution to this linear system. So, the system has infinite solutions. The graph below shows the line resulting from both of the equations in this system. Looking At The Graph Of A Linear System With Infinite SolutionsWhen we graph a linear system with infinite solutions, we will get two lines that overlap. That is, they intersect at every point on the line, since the two equations are equivalent and give us the same line. Let’s take a look at some examples to see how this can happen. Example 1: Graph Of Two Equivalent Equations From A Linear System With Infinite SolutionsLet’s graph the following system of linear equations:
The lines have the same slope (m = 2) and the same y-intercept (b = 4), as you can see in the graph below: Since the slopes are the same and the y-intercepts are the same, the equations represent the same line. So, they will intersect at every point on the line. This means that there are infinite solutions to the linear system we started with. Example 2: Graph Of Two Equivalent Equations From A Linear System With Infinite SolutionsLet’s graph the following system of linear equations:
The lines are horizontal, so they both have the same slope (m = 0). They also have the same y-intercept (b = 4), as you can see in the graph below: Since the slopes are the same and the y-intercepts are the same, the equations represent the same line. So, they will intersect at every point on the line. This means that there are infinite solutions to the linear system we started with. Looking At The Slope & Y-Intercept Of A Linear System With Infinite SolutionsWhen we solve a linear equation for y, we get slope-intercept form. If we do this for both equations in a linear system, we can compare the slope and y-intercept. If the two slopes are the same and the y-intercepts are the same, then the two lines are equivalent, meaning they intersect at all points on the line and there are infinite solutions to the linear system. Let’s take a look at some examples to see how this can happen. Example 1: Comparing Slope & Y-Intercept To Show There Are Infinite Solutions To A System Of Two Linear EquationsLet’s say we have the following system of linear equations:
We will solve for y in both equations to get slope-intercept form, y = mx + b. Solving the first equation for y, we get:
Solving the second equation for y, we get:
So, the two equations in slope-intercept form are:
Since these two equations have the same slope (m = -2) and the same y-intercept (b = 4), we know that they represent the same line. Since the lines intersect at all points on the line, there are infinite solutions to the system. Example 2: Comparing Slope & Y-Intercept To Show There Are Infinite Solutions To A System Of Two Linear EquationsLet’s say we have the following system of linear equations:
We will solve for y in both equations to get slope-intercept form, y = mx + b. Solving the first equation for y, we get:
Solving the second equation for y, we get:
So, the two equations in slope-intercept form are:
Since these two equations have the same slope (m = -2) and the same y-intercept (b = 4), we know that they represent the same line. Since the lines intersect at all points on the line, there are infinite solutions to the system. How To Create A System Of Linear Equations With Infinite SolutionsTo create a system of linear equations with infinite solutions, we can use the following method:
Example: Create A System Of Linear Equations With Infinite SolutionsFirst, we choose any values for a, b, and c that we wish. This gives us our first equation:
Next, we choose a nonzero value of d: d = 4. Now, we multiply both sides of the first equation by d = 4:
Now we have our second equation. Our system of two equations is:
Since the two equations are equivalent, they represent the same line on a graph. So, there are infinite solutions to this system. System Of Linear Equations In Three Variables With Infinite SolutionsA system of equations in 3 variables will have infinite solutions if the planes intersect in an entire line or in an entire plane. The latter case occurs if all three equations are equivalent and represent the same plane. Here is an example of the second case:
Note that the second equation is the first equation multiplied by 2 on both sides. Also note that the third equation is the first equation multiplied by 3 on both sides. Since the equations are all multiples of one another, they are equivalent. That means they all represent the same plane. So, their intersection is the entire plane described by the equation x + y + z = 1. This means that there are infinite solutions to the above system: every point on the plane x + y + z = 1. When Does A System Of Linear Equations Have A Solution?A system of linear equations in two variables has a solution when the two lines intersect in at least one place.
When Does A System Of Linear Equations Have No Solution?A system of two linear equations in two variables has no solution when the two lines are parallel. From an algebra standpoint, this means that we get a false equation when solving the system. Visually, the lines never intersect on a graph, since they have the same slope but different y-intercepts. You can learn more about this case (and some examples) in my article here. ConclusionNow you know when a system of linear equations has infinite solutions. You also know what to look out for in terms of the slope, y-intercept, and graph of lines in these systems. You can learn about systems of linear equations with one solution in my article here and systems of linear equations with no solutions in my article here. You can learn more about slope in this article. You can learn about other equations with infinite solutions here. I hope you found this article helpful. If so, please share it with someone who can use the information. Don’t forget to subscribe to my YouTube channel & get updates on new math videos! ~Jonathon How do you write a system of equations that has infinite solutions?A system of linear equations has infinite solutions when the graphs are the exact same line.
How do you complete an equation so it has infinitely many solutions?If we end up with the same term on both sides of the equal sign, such as 4 = 4 or 4x = 4x, then we have infinite solutions. If we end up with different numbers on either side of the equal sign, as in 4 = 5, then we have no solutions.
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