Prerequisite Skills Review 1.) Simplify: 8r + (-64r) 2.) Solve: 3x + 7(x – 1) = 23 3.) Decide whether the ordered pair (3, -7) is a solution of the equation 5x + y = 8

Solving Systems by Graphing Section 7.1 Solving Systems by Graphing What is a system???? Working with 2 equations at one time: Example: 2x – 3y = 6 X + 5y = -12

What is a system of equations? A system of equations is when you have two or more equations using the same variables. The solution to the system is the point that satisfies ALL of the equations. This point will be an ordered pair. When graphing, you will encounter three possibilities.

Intersecting Lines The point where the lines intersect is your solution. The solution of this graph is (1, 2) (1,2)

infinitely many solutions IDENTIFYING THE NUMBER OF SOLUTIONS NUMBER OF SOLUTIONS OF A LINEAR SYSTEM y x y x y x Lines intersect one solution Lines are parallel no solution Lines coincide infinitely many solutions

Parallel Lines These lines never intersect! Since the lines never cross, there is NO SOLUTION! Parallel lines have the same slope with different y-intercepts.

Coinciding Lines These lines are the same! Since the lines are on top of each other, there are INFINITELY MANY SOLUTIONS! Coinciding lines have the same slope and y-intercepts.

What is the solution of the system graphed below? (2, -2) (-2, 2) No solution Infinitely many solutions

Name the Solution

Name the Solution

Name the Solution

How to Use Graphs to Solve Linear Systems x y Consider the following system: x – y = –1 x + 2y = 5 We must ALWAYS verify that your coordinates actually satisfy both equations. (1 , 2) To do this, we substitute the coordinate (1 , 2) into both equations. x – y = –1 (1) – (2) = –1  x + 2y = 5 (1) + 2(2) = 1 + 4 = 5  Since (1 , 2) makes both equations true, then (1 , 2) is the solution to the system of linear equations.

Solving a system of equations by graphing. Let's summarize! There are 3 steps to solving a system using a graph. Graph using slope and y – intercept or x- and y-intercepts. Be sure to use a ruler and graph paper! Step 1: Graph both equations. This is the solution! LABEL the solution! Step 2: Do the graphs intersect? Substitute the x and y values into both equations to verify the point is a solution to both equations. Step 3: Check your solution.

1) Find the solution to the following system: 2x + y = 4 x - y = 2 Graph both equations. I will graph using x- and y-intercepts (plug in zeros). Graph the ordered pairs. 2x + y = 4 (0, 4) and (2, 0) x – y = 2 (0, -2) and (2, 0)

Graph the equations. 2x + y = 4 (0, 4) and (2, 0) x - y = 2 Where do the lines intersect? (2, 0) 2x + y = 4 x – y = 2

Check your answer! To check your answer, plug the point back into both equations. 2x + y = 4 2(2) + (0) = 4 x - y = 2 (2) – (0) = 2 Nice job…let’s try another!

2) Find the solution to the following system: y = 2x – 3 -2x + y = 1 Graph both equations. Put both equations in slope-intercept or standard form. I’ll do slope-intercept form on this one! y = 2x + 1 Graph using slope and y-intercept

Graph the equations. y = 2x – 3 m = 2 and b = -3 y = 2x + 1 Where do the lines intersect? No solution! Notice that the slopes are the same with different y-intercepts. If you recognize this early, you don’t have to graph them!

y = 2x + 0 & y = -1x + 3 Up 2 and right 1 Down 1 and right 1 (1,2) Slope = -1/1 Y Slope = 2/1 y-intercept= 0 X Up 2 and right 1 Down 1 and right 1 y-intercept= +3 (1,2) The solution is the point they cross at (1,2)

y = x - 3 & y = -3x + 1 Slope = -3/1 Slope = 1/1 y-intercept= -3 The solution is the point they cross at (1,-2)

y =-2x + 4 & y = 2x + 0 Slope = 2/1 Slope = -2/1 y-intercept= 4 The solution is the point they cross at (1,2)

Graphing to Solve a Linear System Solve the following system by graphing: 3x + 6y = 15 –2x + 3y = –3 x y Using the slope intercept form of these equations, we can graph them carefully on graph paper. (3 , 1) Label the solution! Start at the y - intercept, then use the slope. Lastly, we need to verify our solution is correct, by substituting (3 , 1). Since and , then our solution is correct!

Practice – Solving by Graphing y – x = 1  (0,1) and (-1,0) y + x = 3  (0,3) and (3,0) Solution is probably (1,2) … Check it: 2 – 1 = 1 true 2 + 1 = 3 true therefore, (1,2) is the solution (1,2)

Practice – Solving by Graphing Inconsistent: no solutions y = -3x + 5  (0,5) and (3,-4) y = -3x – 2  (0,-2) and (-2,4) They look parallel: No solution Check it: m1 = m2 = -3 Slopes are equal therefore it’s an inconsistent system

Practice – Solving by Graphing Consistent: infinite sol’s 3y – 2x = 6  (0,2) and (-3,0) -12y + 8x = -24  (0,2) and (-3,0) Looks like a dependant system … Check it: divide all terms in the 2nd equation by -4 and it becomes identical to the 1st equation therefore, consistent, dependant system (1,2)

Ex: Check whether the ordered pairs are solns. of the system. x-3y= -5 Ex: Check whether the ordered pairs are solns. of the system. x-3y= -5 -2x+3y=10 (-5,0) -5-3(0)= -5 -5 = -5 -2(-5)+3(0)=10 10=10 Solution (1,4) 1-3(4)= -5 1-12= -5 -11 = -5 *doesn’t work in the 1st eqn, no need to check the 2nd. Not a solution.

Ex: Solve the system graphically. 2x-2y= -8 2x+2y=4 (-1,3)

Ex: Solve the system graphically. 2x+4y=12 x+2y=6 1st eqn: x-int (6,0) y-int (0,3) 2ND eqn: What does this mean? the 2 eqns are for the same line! ¸ many solutions

Ex: Solve graphically: x-y=5 2x-2y=9 1st eqn: x-int (5,0) y-int (0,-5) 2nd eqn: x-int (9/2,0) y-int (0,-9/2) What do you notice about the lines? They are parallel! Go ahead, check the slopes! No solution!

What is the solution of this system? 3x – y = 8 2y = 6x -16 (3, 1) (4, 4) No solution Infinitely many solutions

You Try It Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, determine the solution.

Problem 1 y The two equations in slope-intercept form are: x Plot points for each line. Draw in the lines. These two equations represent the same line. Therefore, this system of equations has infinitely many solutions .

Problem 2 y The two equations in slope-intercept form are: x Plot points for each line. Draw in the lines. This system of equations represents two parallel lines. This system of equations has no solution because these two lines have no points in common.

Problem 3 y The two equations in slope-intercept form are: x Plot points for each line. Draw in the lines. This system of equations represents two intersecting lines. The solution to this system of equations is a single point (3,0) .

Solve a system of two linear equations in two variables graphically. Key Skills Solve a system of two linear equations in two variables graphically. 2 4 –2 6 –4 x y –6 y = 2x  1 y =  1 2 x + 4 solution: (2, 3)

Solve a system of two linear equations in two variables graphically. Key Skills Solve a system of two linear equations in two variables graphically. 2 4 –2 6 –4 x y –6 y + 2x = 2 y + x = 1 solution:≈ (1, 0)

Solve a system of two linear equations in two variables graphically. Key Skills Solve a system of two linear equations in two variables graphically. 2 4 –2 6 –4 x y –6 y = 2x + 2 y = 2x + 4 No solution, why? Because the 2 lines have the same slope.

Solve a system of two linear equations in two variables graphically. Key Skills TRY THIS Solve a system of two linear equations in two variables graphically. 2 4 –2 6 –4 x y –6 y = 3x + 2 y =  1 3 x - 2 solution:≈ (-3, -1)

Solve a system of two linear equations in two variables graphically. Key Skills TRY THIS Solve a system of two linear equations in two variables graphically. 2 4 –2 6 –4 x y –6 2x + 3y = -12 4x – 4y = 4 solution:≈ (-1.5, -3)

Consider the System BACK

Graph each system to find the solution: 1.) x + y = -2 2x – 3y = -9 2.) x + y = 4 2x + y = 5 3.) x – y = 5 2x + 3y = 0 4.) y = x + 2 y = -x – 4 5.) x = -2 y = 5 (-3, 1) (1, 3) (????) (????) (-2, 5)

Check whether the ordered pair is a solution of the system: 1.) 3x + 2y = 4 (2, -1) -x + 3y = -5 2.) 2x + y = 3 (1, 1) or (0, 3) x – 2y = -1 3.) x – y = 3 (-5, -2) or (4, 1) 3x – y = 11