Warm Up Graph the following lines: y = 2x – 3 b. 4x + 8y = 16 Solve the equation -3(x + 5) = 2x + 10 If two lines are parallel, how many times do they intersect? How about if they are perpendicular?

Systems of Equations!

So what is a system of equations and how will it help us? For awhile, we’ve been working with linear equations. Linear equations were equations that had an x and a y. (Like y = 3x + 4 or 2x – 7y = 30) A system of equations just means that we have two or morelinear equations that we are trying to solve to find the values of x and y.

So what is a system of equations and how will it help us? The good news: When we have two equations that both have x and y, we can figure out EXACTLY what x and y are. (Ex) 4x + 3y = 20 and 5x + 2y = 18 x = 2 and y = 4

But how do we solve these systems? When we had one equation, we solved it by using inverse operations. (ex) 5x + 2 = 12 With two equations and two variables, we have 3 methods to choose from: Graphing Substitution Elimination

When we graphed a line, what did that tell us about the equation? On this graph, we have the line y = 2x – 3 This line tells us all of the combinations of numbers that x and y could be that would make our equation true. Let’s look at a couple of points to see what that means.

So what happens when we put two lines (or two equations) on the same graph? y = 3/2x + 1 y = 3x – 3 Does this graph show us the value of x and y that make BOTH equations true? Where?

SO, on a graph… The solution to a system of equations is the point where our two lines intersect. Why? Because each line shows us all of the x and y values that make its equation true So, the spot where BOTH of them are true is where the two lines touch each other

For these two equations, what is our solution? y = 3/2x + 1 y = 3x – 3 At the point where the lines intersect, x = y = We can also write our answer as an ordered pair: ( , )

What are the solutions of the systems below?

Parallel Lines If two lines are parallel, then the system has NO SOLUTION because the lines never intersect.

Same Line y = ½ x y – 3 = ½ (x – 6) When two equations give you the same line on a graph, your system has infinitely many solutions,because there are and endless combination of x and y values that will make the equation true.

When solving a system by graphing: Example What is the solution to the system below: When solving a system by graphing: Make sure each equation is in slope intercept form (solve for y) Use your calculator to graph both lines. Determine the intersection.

Calculator Steps Put equation 1 into Y1 = Put equation 2 into Y2 = Hit graph, and make sure you can see where the lines intersect on your screen. 2nd Calc  Intersect (5) Hit enter 3 times until you see “x = “ and “ y = “ at the bottom of your screen.

You Try these two.

Example 2 For this one, we need to…

You Try these two.

Practice time! Use your systems of equations solving skills to answer the following riddle: What do they put on a criminal pig?

Optional Word Problem At the North Carolina fair, it costs $7.00 to get in and $1.00 per ride. At the South Carolina fair, it costs $5.00 to get in and $1.50 per ride. If one person went to each fair, how many rides would they ride for their cost to be the same? And what would their total cost be?

You Try Tar Heel Taxi charges a flat rate of $2.50 and and $0.50 per quarter mile. Blue Devil Taxi charges a flat rate of $3.00 and $0.40 per quarter mile. For what distance would the two charge the same amount? And what would that charge be? Bonus: If you knew your taxi ride was going to be 2 miles, which company would you choose? Why?

Homework Record Breaker worksheet