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• If given a solution for an equation, you can plug in the solution’s number to see if the numbers match your equation. Ex: Plug in the solution (2,1) to the given equations: • 2x-3y=1 2(2) – 3(1)=1 1=1 • x+y=3 2+1=3 3=3 Checking Solutions of a Linear System Try on your own: Ex: (-2 , -8) 2x – y=52 9x – y=-10 2(-2)–(-8)=52 -4+8=52 ; -4≠52 (-2,-8) Isn’t a solution to the equation 9(-2)—(-8)=-10 -18+8=-10 ; -10 =-10 (-2,-8) Is a solution to the equation Ex: x+2y=9 -x+6y=-1 1. Put the two equations into slope-intercept form (y=mx+b) 2y=-x+9 6y=x-1 y=-x/2 + 9/2 y=x/6 -1/6 2. Graph the new equations. 3. When the two equations are graphed, a point of intersection is made. For this example, (7,1) is the point of intersection. 4. Plug in the point of intersection into the original equations to see if the solution works. • You perform the same steps as before, however the graphs will look different. • Equations with infinitely many solutions will share the same line on the graph. • Equations with no solutions will form parallel lines on a graph. Ex: 3x-2y=6 6x-4y=12 Y=3x/2-3 Y= 3x/2-3 Since the equations share a line, each point on the line is a solution. Therefore there is an infinite amount of solutions. Ex: 3x – 2y=6 3x-2y=2 Y= 3x/2 – 3 Y= 3x/2 – 1 Since the lines are II there is no point of intersection, therefore there is no solution to the equations. There are 2 methods to solving a linear system algebraically: Substitution Method 1. Solve for 1 of the variables in 1 of the equations 2. Substitute the expression from step 1 into the second equation to solve for the other variable 3. Substitute the value from step 2 into the revised equation from step 1 and solve. Ex: 3x+4y=-4 ; x+2y=2 1. Solve for x in one the equations: x= -2y+2 2. Substitute the expression into the other equation and solve for y: 3(-2y+2) + 4y=-4 -6y+6+4y=-4 -2y+6=-4 -2y=-10 y=5 3. Substitute y into your revised equation from step 1. x= -2(5)+2 x=-10+2 x=-8 y=5 (-8,5) *Check to see if your solution is correct by plugging in the numbers to your original equations. The second method is: Linear Combination: Ex: 2x-4y=13 4x-5y=8 1. Multiply 2x-4y=13 by -2. By doing this, both coefficents in the equations will be 4, only their signs will differ: -4x+8y=-26 4x-5y=8 2. Add these two equations together: The coefficents -4 and 4 cancel out when added together. -4x+8y=-26 3y=-18 y=-6 4x -5y=8 3. Substitute y into the original equation to solve for x: 2x-4(-6)=13 2x+24=13 2x= -11 x= -11/2 (-11/2, -6) is your solution *Check to see if your solution is correct by plugging the numbers into your original equations • Use the same steps in the linear combination method. After multiplying the first equation by 2, add the two equations together: This equation has infinitely many 14x+ -14x + 4y-4y =0 0=0 solutions After multiplying the first equation by 3, add the two equations together: This equation has no 18x + -18x +-3y + 3y=-2 0≠-2 solutions. Try It On Your Own Graph: 3x+4y = -10 -7x+ -y = -10 Y=-3x/4 – 5/2 Y= -7x+ 10 (2,-4) Is the point of intersection. 3(2) + 4 (-4) = -10 6 + -16= -10 -10=-10 -7 (2) – (-4) = -10 -14+ 4 = -10 -10=-10 How many solutions are there to this equation? One Try It On Your Own Graph: 7x + 2y= 16 -21x + 6y= 24 Y= -7x/2 + 8 Y= -7x/2 - 4 How many solutions are there to the equations? There are no solutions to the equations Try It On Your Own Solve Using the Substitution Method: Ex: x+ 2y =2 7x – 3y = -20 X= -2y + 2 7(-2y + 2) -3y=-20 -14y + 14 – 3y =-20 -17y=-34 Y=-34/-17 Y=2 X=-2(2) + 2 X= -2 Try It On Your Own Solve by Using the Linear Combination Method Ex: 3x – y = 4 9x – 3y = 12 9x – 3y = 12 -9x + 3y = -12 -9x + 3y = -12 -9x + 3y = -12 0=0 How many solutions are there for these equations? Infinitely many solutions Try It On Your Own Solve by Using the Linear Combination Method Ex: 3x – y =7 2x + 3y =1 9x – 3y = 21 2x + 3y =1 9x – 3y = 21 2x + 3y =1 11x=22 x=2 3 (2) – y =7 6 – y =7 -y = 1 Y= -1 How many solutions are there to the equations? One