Today, we're going to talk about systems of equations. A system of equations is just a set of two or more equations that have the same variables. And you consider the equations at the same time when solving.
So for example, I have the equations y equals 3x plus 1 and y equals 10 x minus 3. They both have the same variables, x and y. And we're going to solve them at the same time for x and y. So we're going to do some examples to show you how to solve a system of equations and what the solution looks like.
So when we're finding the solution to a system of equations, we must find a solution that satisfies all of the equations within the system. So for example, if I were to take just my first equation, and let's say that I thought the solution for my x variable was x is equal to 3, I could find the value of my y variable by substituting 3 into my equation and simplifying.
So doing that, I would have y is equal 4 times 3 minus 5. So simplifying this I find 4 times 3 is 12. And 12 minus 5 is going to give me 7. So if my x variable is equal to 3, I found that my y variable is going to be equal to 7.
But we know that this is only going to be a solution to my system of equations if those two values of x and y also satisfy my second equation. So I can test that by substituting my two values into the equation to see if it yields a true statement.
So if I substitute 7 in for y, into my second equation, and 3 in for x, I want to see if this statement is true. So simplifying negative 6 times 3 will give me negative 18. Negative 18 plus 15 will give me negative 3. And so I can see that negative 3 is not equal to seven. So the solution x equals 3 and y equals 7 is not a solution to our system of equations because it does not satisfy. It's not a solution to all of the equations in the system.
So let's look at what it means to be the solution to a system of equations by looking at the graph. So the solution of two lines that represent a system of equations can be found by looking at the intersection point of those two lines. So here, since these two lines intersect at 2, negative 2, I know that the solution to my system of equations represented by these two lines will be x equals 2 and y equals negative 2. And I can verify that by looking at my equations and evaluating both equations with an x value of 2 and a y value of negative 2 to see that it doesn't yield a true statement.
So if I look at my first equation, y is equal to negative 5 over 2x plus 3. Again, I'm going to substitute in my values for x and y. So x is equal to 2 and y is equal to negative 2. So negative 2 equals negative 5 over 2 times 2 plus 3. And I want to show that this is the solution by first looking if the yields a true statement with our first equation.
So simplifying this, negative 5 over 2 times 2 is just going to give me a negative 5. And negative 5 plus 3 does indeed equal negative 2. So I can see that the values of 2 for x and negative 2 for y does satisfy my first equation.
But again, I need to make sure that it satisfies all the equations in my system. So I also am going to test this for my second equation, which is y is equal to x minus 4. So again, substituting in my values for x and y, negative 2 for y and positive 2 for x. Simplifying this, 2 minus 4 is just going to give me negative 2. And so I again can see that this intersection point, where x is 2 and y is negative 2, satisfies my second equation. So that means that 2, negative 2 is the solution to my system of equations.
So here I have two graphs of lines that represent a system of two equations. So we also notice that these two lines are parallel to each other, which means that they have the same slope. And because these two lines are parallel to each other, they are never going to intersect. So because they don't intersect, there is no solution to this system of equations, because without an intersection point, there are no x and y values that are going to satisfy the system of these two equations.
So lastly, we have a graph of two identical lines, one in green and one in red. And these two lines are identical, meaning you can think of them as being on top of each other, or sharing all of their coordinate pairs, all of their x and y values are the same. Which also means that they intersect at every point along both lines.
So these two lines have an infinite number of intersection points, which means that the solution to the system of equations is infinite number of solutions. So any time that you have a system of equations where all of the equations are identical, then the solution to that system is going to be infinite number of solutions.
So let's go over our key points from today. As usual, make sure you have them in your notes if you don't already so you can refer to them later.
The solution to a system of equations is a solution to all equations in the system. On a graph, the solution to a system of equations is the intersection point of the lines on the graph. And there's no solution to a system of equations represented by lines that are parallel because the lines will never intersect. But there are an infinite number of solutions to a system of equations represented by lines that are identical because the lines share all points.
So I hope that these key points and examples helped you understand a little bit more about systems of equations. Keep using your notes and keep on practicing. And soon you'll be a pro. Thanks for watching.
two or more equations with the same variable, considered at the same time