Lesson 3-1: Solving Linear Systems by Graphing

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1 For the past several weeks we ve been working with linear equations. We ve learned how to graph them and the three main forms they can take. Today we re going to begin considering what happens when we take two or more linear equations together at the same time. Linear Systems When we consider two or more linear equations together at the same time, it is called a linear system. In other words, it is a system (more than one) of linear equations. Perhaps the most interesting thing to look at with linear systems is where the lines intersect (if they do at all). For instance, if we were running a business and had tracked our costs and expenses, we d likely have come up with an equation that predicted what our expenses will be over time. It would also be likely that we d of tracked our profits (hey, everyone loves to make money right?) and would have an equation that predicted our profits over time. If we graphed those two lines, what do you think the intersection of those lines would mean? It would be where our expenses matched our profits right? That would be what we call our breakeven point. After that point, we d be making money. Before then, we d have been losing money. In a system of linear equations, the intersection of the lines is what we refer to as the solution of the linear system. It is the coordinate point (x, y) of the intersection. How do we find the intersection linear system solution? Today and tomorrow we will be learning three ways of finding the solution to a linear system. To give you an idea of where we re going, here are the three ways: 1. Graph the lines, visually locate the intersection: only approximates the solution. 2. Substitution: solve one equation for a variable, substitute into the other. 3. Combination: add the two equations together to eliminate one variable. We will work on the latter two tomorrow. They are the most precise and are algebraic methods. Today we will work on finding the solution by graphing. Finding the intersection by graphing This is pretty straight-forward. Just graph the two lines and find where they intersect. Read off the coordinates and your done right? Not really. How do you know if those coordinates are right? You are drawing the graph; how accurate were you? What if you are off by a tenth? What can you do to decide if the coordinates you read off are correct? Page 1 of 5

2 What does the intersection of two lines mean? It means the point they both go through. Another way of looking at that is that is the only x value for which both equations have the same y value. Plug x into both equations, you should get the same answer from each. That s our answer! Once we read off the coordinates, just plug them into each equation to verify we get the same answer! Easy! What you will need to do today Today s assignment will require you to be able to do the following: 1. Match a system of linear equations with its graph. 2. Determine the number of solutions the system of linear equations has. 3. Decide if a coordinate pair is a solution for a given system of linear equations. 4. Sketch a graph of a system of linear equations to estimate and check the solution. Find the solution of y = 2x + 2 and y = -2x 2 1. Graph both lines. Looking at the graph, it looks like they cross at (-1, 0). Check it: y 2x 2 y 2x 2 0 2( 1) 2 0 2( 1) Both check out, so (-1, 0) is the solution for y = 2x + 2 and y = -2x 2. Page 2 of 5

3 Find the solution of -x + 5y = 5 and 2x 10y = Graph both lines this time they re in standard form x & y intercepts Hmm, a bit of a problem isn t there. It doesn t look like these two lines intersect. Is there an equation form that would make it very easy to compare the two equations? Yup, the slope-intercept form y = mx + b. Solve both equations for y to get them into slope-intercept form: -x + 5y = 5 y = 1 / 5 x + 1 2x 10y = 30 y = 1 / 5 x 3 Great! We can now see that both have the same slope ( 1 / 5 ) but different y-intercepts. Since they cross the y axis at different points this means they are parallel and will never cross. This system of linear equations does not have a solution. Page 3 of 5

4 Find the solution of -2x + y = 3 and 6x 3y = Graph both lines again, they re in standard form x & y intercepts Ok, this was a trick right? They re the same line aren t they? Yup is there a way we can determine that by looking at the equations? Sure is! What is the line form that allows us to directly compare two equations? The slope-intercept form; put both equations into slope-intercept form: -2x + y = 3 y = 2x + 3 6x 3y = -9 y = 2x + 3 Now it is easy to see they are actually the same line! Next question is how many solutions does this system of linear equations have? The number of solutions is the same as the number of times the lines intersect. An intersection is a point both lines have in common. So, how many points do these two equations / lines have in common? All of them, or put another way an infinite number. This system of linear equations has an infinite number of solutions. Page 4 of 5

5 Summary The number of solutions a system of linear equations can have: 1. One: the equations represent different lines that intersect at one point. 2. None: the equations represent parallel lines. 3. Infinite: the equations represent the same line. How to find the solution of a system of linear equations by graphing: 1. Graph both lines. 3. Plug back into both equations and check. How to verify if the lines are parallel or are the same: 1. Put both equations into slope-intercept form (solve both for y). 2. If they have the same slope but different y-intercepts, they are parallel. 3. If they have the same slope and same y-intercepts, they are the same line. Page 5 of 5

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