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Transcript
Name _____________________________ Algebra I Date ___________ Period ___________ Systems of Equations Let’s Review: How do we graph an equation? 1) Place the equation in y-intercept form. 2) Graph the y-intercept. 3) Use the slope to get a second point. 4) Connect the points. Graph: -2x+3y= -18 Defining a System of Equations A grouping of ______________________________, containing one or more variables. x+y=2 2x + y = 5 2y = x + 2 y = 5x - 7 6x - y = 5 1 Name _____________________________ Date ___________ Algebra I Period ___________ How do we “solve” a system of equations??? By finding the point where two or more equations ________________. x+y=6 y = 2x 6 We also need to verify that our solution satisfies both equations. Point of intersection 4 2 1 2 Determining whether a solution works… 1) Plug the point into both equations 2) Solve each equation to make sure both sides are = Example: #1) Is (2, -3) a possible answer for 4x+8y=-16 and 5x+ 3y=2? #2) Is (-1, 3) a solution for 5x+4y=7 and 2x-8y=-26? Graphing to Solve a Linear System Let's summarize! There are 4 steps to solving a linear system using a graph. 2 Name _____________________________ Algebra I Step 1: Put both equations in y intercept form. Date ___________ Period ___________ Solve both equations for y, so that each equation looks like y = mx + b. Step 2: Graph both equations on the same coordinate plane. Use the slope and y - intercept for each equation in step 1. Be sure to use a ruler and graph paper! Step 3: Estimate where the graphs intersect. This is the solution! LABEL the solution! Step 4: Check to make sure your solution makes both equations true. Substitute the x and y values into both equations to verify the point is a solution to both equations. Solve the following system of equations by graphing. Example #1) 2x + 2y = 3 x – 4y = -1 Example #2) 2x+3y=18 -4x+2y=-4 3 Name _____________________________ Algebra I Example #3) 5x+2y=10 15x+6y=30 Date ___________ Period ___________ Example #4) -2x+y=3 -2x+y=1 Types of Solutions No Solution: when lines of a graph are _______________ since they do ________________________, there is no solution also called an ________________________ Infinite Solutions: a pair of equations that have the same ____________________ and ____________________. also call a ___________________________ 4 Name _____________________________ Algebra I Unique Solution: the lines of two equations ___________________ also called an ____________________ Date ___________ Period ___________ Determine whether the following have a unique, no solution, or infinite solutions by looking at the slope and y-intercepts #1) #2) #3) x2y + x = 8 y = -6x + 8 y = 2x + 4 y + 6x = 8 5y = 10 -5y = -x +6 Solving Equations by Graphing on the Calculator 1) Make sure both equations are in y-intercept form 2) Hit y= 3) Input both equations 4) Hit graph 5) Hit 2nd Calc 6) Choose #5: Intersect 7) Hit Enter until it gives you the intersection #1) y=7+2.5x y=35.9-6x #2) y=25+30x y= 15+32x #3) y=4x-5.5 y= -3x+5 #4) 2x+y=9 3x+y=16.3 5 Name _____________________________ Algebra I Date ___________ Period ___________ 2 Methods for Solving Algebraically 1. Substitution Method: (used mostly when one of the equations has a variable with a __________________________________________) 2. Elimination Method Substitution Method 1. Solve one of the given equations for one of the variables. (whichever is the easiest to solve for) 2. Substitute the expression from step 1 into the other equation and solve for the remaining variable. 3. Substitute the value from step 2 into the original equation and solve for the 2nd variable. 4. Check your solution. 5. Write the solution as an ordered pair (x,y). Ex #1: Solve using substitution method 3x-y=13 2x+2y= -10 1. Solve the 1st eqn for y. 3x-y=13 -y= -3x+13 y=3x-13 2. Now substitute 3x-13 in for the y in the 2nd equation. 2x+2(3x-13)= -10 Now, solve for x. 2x+6x-26= -10 8x=16 x=2 3. Now substitute the 2 in for x in for the equation from step 1. y=3(2)-13 y=6-13 y=-7 4. Solution: (2,-7) 6 Name _____________________________ Algebra I Date ___________ Period ___________ 5. Plug in to check solution. #2) y 2x 2 2x 3y 10 Step 1: Solve one equation for one variable. Step 2: Substitute the expression from step one into the other equation and solve the equation. Step 3: Substitute back into either original equation to find the value of the other variable. Step 4: Check solution. 7 Name _____________________________ Algebra I #3) y = 4x 3x + y = -21 Date ___________ Period ___________ #4) x + y = 10 5x – y = 2 8 Name _____________________________ Algebra I 2a 3b 7 #5) Date ___________ Period ___________ 2a b 5 Elimination Method 1. Multiply one or both equations by a real number so that when the equations are added together one variable will cancel out. 2. Add the 2 equations together. Solve for the remaining variable. 3. Substitute the value form step 2 into one of the original equations and solve for the other variable. 4. Write the solution as an ordered pair (x,y). Ex #1): 2x-6y=19 -3x+2y=10 1. Multiply the entire 2nd eqn. by 3 so that the y’s will cancel. 2x-6y=19 -9x+6y=30 2. Now add the 2 equations. -7x=49 and solve for the variable. x=-7 3. Substitute the -7 in for x in one of the original equations. 9 Name _____________________________ Algebra I 2(-7)-6y=19 -14-6y=19 -6y=33 y= -11/2 4. Now write as an ordered pair. (-7, -11/2) Date ___________ Period ___________ 5. Plug into both equations to check. #2) x+y=7 x-y=3 #3) 2x - y = - 8 2x + 2y = 16 10 Name _____________________________ Algebra I #4) 4x+5t=22 5x-t=13 Date ___________ Period ___________ #5) 4x + 2y = 24 - 3x + 6y = - 63 11 Name _____________________________ Algebra I #6) 9x-3y=15 -3x+y= -5 Date ___________ Period ___________ Both equations are for the same line! ______________________ Means any point on the line is a solution. #7) 6x-4y=14 -3x+2y=7 It means the 2 lines are parallel._________________________________ Since the lines do not intersect, they have no points in common. 12 Name _____________________________ Algebra I #8) 3x-2y=4 2x+y=4 Date ___________ Period ___________ #9) 3c-8d=7 c+2d=-7 #10) 2c-7d=41 6c+5d=-7 13 Name _____________________________ Algebra I #11) 2x + 4y = 50 x - 8y = - 65 Date ___________ Period ___________ #12) 5x - 5y = - 25 3x - 15y = - 39 14