[MUSIC PLAYING] Let's look at our objectives for today. We'll start by introducing a system of equations. We'll then look at how to graph a system of equations. And finally, we'll do some examples finding the intersection point from two lines.
Let's start by introducing a system of equations. Systems of equations have applications in economics, physics, and engineering. A system of linear equations consists of two or more equations with the same variables considered at the same time. This means that the equation in a system are graphed together on a single coordinate plane. The solution to this system of equations is the solution that satisfies all individual equations in the system.
For example, here we have a system of equations and their graphs. The point of intersection of the lines represents the solution to the two equations of the lines. The intersection point represents an x value and y value that can be substituted into each equation in the system to yield a true statement.
Here is another example. We have the system of equations y equals 3x minus 5 and y equals 3x plus 2. These two lines are parallel-- meaning they do not have an intersection point, which means there is no solution to the system of equations.
Now, let's see how we can solve a system of equations by graphing. A company is analyzing supply and demand data for their product. The system of equations represents the price of the product in relation to the quantity supplied and the quantity demanded. We can represent the situation with a system of equations, y equals negative 1 1/2 x plus 60, which represents the demand of the product, and y equals 3 1/2 x, which represents the supply for the product.
To solve, we will graph both equations and find the intersection point. In the first equation, the y-intercept is 60. So we have a point at 0, 60. We then use our slope, which is negative 1/2. Since our graph uses intervals of 5, we moved down 5 and over 10 from the y-intercept to place our second point. Now, we connect both points with a line.
In the second equation, the y-intercept is 0 because there is no written b value. So we have a point at the origin 0, 0. We then use our slope, which is 3 over 2. So from our y-intercept, we move up 3 and over 2 to place our second point.
Now, we connect both points with the line. We see that our lines intersect at the point 30, 45, which means that x equals 30 and y equals 45 is the solution to our system. This is where the supply and demand are equal, and the company will maximize their profits by setting the price at $30 and selling a quantity of 45 units. We can verify this by substituting the solution into both equations to see if they yield true statements. Doing this in the first equation gives us 45 equals negative 1/2 times 30 plus 60.
Simplifying on the right side gives us negative 15 plus 60, which is 45. Substituting our values for x and y into our second equation gives us 45 equals 3/2 times 30. Simplifying the right side gives us 45, so 45 equals 45. Therefore, our solution of x equals 30 and y equals 45 is correct.
Let's look at our last example. A company is testing a new product. While the product is running, they measure the temperature every minute. They need to make sure the temperature stays below 85 degrees. We can represent this situation with a system of equations. The first equation, y equals 2x plus 45, represents the temperature of the machine, and y equals 85 represents the maximum temperature that can be reached.
To solve, we will graph both equations and find the intersection point. In the first equation, the y-intercept is 45. So we have our first point at 0, 45.
We then see our slope is 2, which is the same as 10 over 5. So on our graph, we can move up to 10 and over 5 to place our second point. Now, we connect both points with the line.
The second equation is a horizontal line that goes through 85 on the y-axis, so we plot a point at 0, 85 and draw a horizontal line through the point. We see that these two lines intersect at the point 20, 85, which means that x equals 20 and y equals 85 is the solution to our system. That also means that it will take 20 minutes for the temperature to reach 85 degrees.
We can verify this by substituting the solution into both equations to see if they yield true statements. Doing this in the first equation gives us 85 equals 2 times 20 plus 45. Simplifying on the right side gives us 40 plus 45, which is 85.
For our second equation, we substitute 85 in for y and have a true statement 85 equals 85. So our solution x equals 20, y equals 85 is correct.
Let's go over our key points from today. Make sure you get these in your notes so you can refer to them later. A system of linear equations consists of two or more equations with the same variables considered at the same time. The solution to a system of equations is a solution that satisfies all individual equations in the system.
On a graph, the point of intersection of the lines represents the solution to the system of the equations. The intersection point represents an x value and a y value that can be substituted into each equation in the system to yield true statements. And lines that are parallel do not have an intersection point, which means there is no solution to the system of equations.
So I hope that these key points and examples helped you understand a little bit more about finding the intersection point of two lines. 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 variables, considered at the same time.