Test 1 Results Average class score after partial credit Commonly missed questions Grade Scale If you got less than 70 on Test 1 make sure to go over your quiz with me or a TA sometime today or tomorrow to help you prepare for tomorrow s test Now please CLOSE YOUR LAPTOPS and turn off and put away your cell phones Sample Problems Page Link Dr Bruce Johnston Section 4 1A Solving Systems of Equations in Two Variables A system of linear equations consists of two or more linear equations This section focuses on only two equations at a time The solution of a system of linear equations in two variables is any ordered pair that solves both of the linear equations What does this look like on a graph The SOLUTION to a system of two linear equations is the intersection if any of the two lines There are only three possible solution scenarios 1 The lines intersect in a single point so the answer is one ordered pair 2 The lines don t intersect at all i e they are parallel so the answer is no solution 3 The two lines are identical i e coincident so there are infinitely many solutions all of the points that fall on that line To be a SOLUTION of a system of equations an ordered pair must result in true statements for BOTH equations when the values for x y are plugged into them If either one or both gives a false statement the ordered pair is NOT a solution of the system Determine whether the given point is a solution of the following system point 3 1 system x y 4 and 2x 10y 4 Plug the values into the equations First equation 3 1 4 true Second equation 2 3 10 1 6 10 4 true Since the point 3 1 produces a true statement in both equations it is a solution Determine whether the given point is a solution of the following system point 4 2 system 2x 5y 2 and 3x 4y 4 Plug the values into the equations First equation 2 4 5 2 8 10 2 true Second equation 3 4 4 2 12 8 20 4 false Since the point 4 2 produces a true statement in only one equation it is NOT a solution Problem from today s homework Note that our chances of guessing the right coordinates for a solution just by looking at the two equations are not very good Since a solution of a system of equations is a solution common to both equations it would also be a point common to the graphs of both equations So one way to find the solution of a system of 2 linear equations is to graph the equations and see where the lines intersect You can use any of the techniques from Chapter 3 to graph the two lines e g solving each equation for y and using the slope and intercept or making a table of x and y values for each equation and plotting the ordered pairs Graphing is the first of three methods that we will be studying in this section to find a solution for a system of equations The other two methods we will be using are Substitution method also covered today Addition or elimination method in Sec 4 1B Note Graph Paper Click on the News Bulletins button to find a site that allows you to print free graph paper If you want to be able to draw accurate graphs but you don t want to buy a whole pack of graph paper for one assignment go to this web site and print a couple pages of graph paper for free You don t have to do this graphing by hand on plain paper is fine but sometimes it s easier to see the solutions if you can plot your points carefully on real graph paper instead of a hand drawn graph grid http www printfree com Office forms GraphPaper2 htm y Solve the following system of equations by graphing 5 5 2 4 6 6 1 3 2x y 6 and x 3y 10 3 0 First graph 2x y 6 Second graph x 3y 10 The lines APPEAR to intersect at 4 2 4 2 0 6 x Although the solution to the system of equations appears to be 4 2 you still need to check the answer by substituting x 4 and y 2 into the two equations First equation 2 4 2 8 2 6 true Second equation 4 3 2 4 6 10 true The point 4 2 checks so it is the solution of the system Problem from today s homework y Solve the following system of equations by graphing x 3y 6 and 3x 9y 9 6 4 0 2 3 0 6 0 First graph x 3y 6 Second graph 3x 9y 9 The lines APPEAR to be parallel 0 1 6 1 x Although the lines appear to be parallel you still need to check that they have the same slope You can do this by solving for y First equation x 3y 6 3y x 6 add x to both sides 1 y 3x 2 divide both sides by 3 Second equation 3x 9y 9 9y 3x 9 1 y 3x 1 subtract 3x from both sides divide both sides by 9 1 3 Both lines have a slope of since they have different y intercepts they are parallel and do not intersect Hence there is no solution to the system y Solve the following system of equations by graphing x 3y 1 and 2x 6y 2 5 2 1 0 4 1 7 2 First graph x 3y 1 Second graph 2x 6y 2 The lines APPEAR to be identical 2 1 x Although the lines appear to be identical you still need to check that they are identical equations You can do this by solving for y First equation x 3y 1 3y x 1 1 y 3x add 1 to both sides 1 3 divide both sides by 3 Second equation 2x 6y 2 6y 2x 2 1 1 y 3x 3 subtract 2x from both sides divide both sides by 6 The two equations are identical so the graphs must be identical There are an infinite number of solutions to the system all the points on the line y 1 3 x 1 3 Watch out for graphing problems 5 and 6 when the y intercept is not an integer Solving the first equation for y gives y 2x 2 which can be graphed by starting at the y intercept of 0 2 and then using the slope 2 1 to produce a second point Solving the second equation for y gives y 1 3 x 13 3 The graphing tool will not allow you to plot the yintercept 0 13 3 To use the graphing tool you must plot two points by selecting integer values for x that also produce integer values for y Substitution Method A second method that can be used to solve systems of equations is called the …