SOLVING SYSTEMS OF EQUATIONS

Transcription

1 SOLVING SYSTEMS OF EQUATIONS GRAPHING System of Equations: 2 linear equations that we try to solve at the same time. An ordered pair is a solution to a system if it makes BOTH equations true. Steps to Solve a System by Graphing: Worked Out Example: x + y = 5 3x y = 3 1. Solve both equations for y. x + y = 5 3x y = 3 x x 3x 3x y = 5 x y = 3 3x 1 1 y = 3 + 3x 2. Use your graphing calculator. Hit the Y= button Enter one of your equations into Y 1 and the other into Y 2 (Use PARENTHESIS!!) Ex: If you have y = 2 x + 3 you would enter it in your 3 4 calculator like this: (2/3)x+(3/4) The X,T,,n button is how you Enter an X into your calculator. Hit the Graph button. It should look like this: 1

2 3. Now, hit the 2 nd button and then the Graph button. This will bring you to the table which looks like this: The x-value is on the left and the Y 1 column has the respective y value for the first equation, and the Y 2 column has the y-value for the second equation. You can use the up and down arrows to go through the table. Use this table, take a few points and sketch a graph for yourself. *When you are asked to Solve by graphing you MUST have the lines graphed as part of your answer. 4. YOUR ANSWER: will be the ordered pair (point) where the two lines intersect. You have two ways to find this point. Option 1: Use your table. When the y-values are the same for ONE x-value then that x-value and the same y-value will be your answer. For the example: At the x-value of 2, both of the y-values are 3 so your answer is (2,3) Option 2: Use the Trace Feature of your calculator (When the table is not helpful). When looking at your graphed functions. Hit the Trace button A flashing dot will pop up and you can use the left and right arrow keys to move the point until it is on the intersection point. At that time the x= and y= at the bottom of the graph will tell you the x and y values for that intersection point: Your Answer. If it is a long decimal, round to 2 decimal places. 5. One important note about solving by graphing, is that if you have to use the trace feature it is not always accurate, and therefore you should only rely on it to check your answers. 2

3 Use the above steps to solve the following systems of equations by graphing. Example 1: 2x + 3y = 6 2x + y = 2 Example 2: x + 2 = y x 1 = y Example 3: x + 2y = 5 2x + 4y = 2 3

4 **Remember that when solving, you can also have either: INFINITE SOLUTIONS or NO SOLUTION. Example 4: y = x + 1 y = x + 5 What type of solution does this system have? This system has Because Example 5: x + y = 5 2x + 2y = 10 What types of solution does this system have? This system has Because 4

5 SOLVING SYSTEMS OF EQUATIONS SUBSTITUTION System of Equations: 2 linear equations that we try to solve at the same time. An ordered pair is a solution to a system if it makes BOTH equations true. Steps to Solve Systems by Substitution: Worked Out Example: x = 3 + 2y 5x 4y = 9 1. Solve ONE equation for one variable if necessary The first equation is (It is easiest to solve for a variable that has a already solved for one coefficient of 1). variable: x = 3 + 2y 2. Substitute this expression in for the variable x = 3 + 2y 5x 4y = 9 In the other equation. (This should leave you With an equation with ONE variable). 5( 3 + 2y) 4y = 9 3. Solve the new equation for the variable. 5( 3 + 2y) 4y = y 4y = y = 9 (Combine like terms) (Get y alone) 6y = y = 4 (Get y alone, divide by 6) 4. Plug this answer back into the other x = 3 + 2y for y = 4 Equation to solve for the remaining x = 3 + 2(4) Variable. x = x = 5 5. Write your solution as an ordered pair. Answer: (5, 4) Check your answer. 5 = 3 + 2(4) 5(5) 4(4) = 9 5

6 Use the above steps to solve the following systems of equations by substitution. Example 1: 3x + 2y = 1 x y = 3 Example 2: y = 7x + 1 7x + 3y = 3 Example 3: 5x + 2y = 11 x + y = 4 6

7 **Remember that when solving, you can also have either: INFINITE SOLUTIONS or NO SOLUTION. Example 4: y = 3x 2 15x 5y = 10 What type of solution does this system have? This systems has Because. Example 5: x + y = 4 2x + 2y = 6 What type of solution does this system have? This systems has Because. 7

8 SOLVING SYSTEMS OF EQUATIONS ADDITION (ELIMINATION) System of Equations: 2 linear equations that we try to solve at the same time. An ordered pair is a solution to a system if it makes BOTH equations true. Steps to Solve Systems by Addition (Elimination): Worked Out Example: Solve using the addition method. 3x + 2y = 48 9x 8y = If necessary, rewrite BOTH equations Both equations are in this form. In the form ax + by = c 3x + 2y = 48 and 9x 8y = 24 Where a, b, and c are just numbers. 2. If necessary, multiply the equation(s) by a 3x + 2y = 48 constant (number) so that the coefficients 9x 8y = 24 (numbers in front of the variables) of ONE VARIABLE, differ only in sign. Option 1: I can multiply the top by 3 **Option 1: Both numbers in front of x are the SAME, but one is negative and one is positive. Then I would have 3(3x + 2y = 48) ( 3)(3x) + ( 3)(2y) = ( 3)(48) 9x 6y = 144 Now, compared with the second, the numbers in front of x are the same, but one is negative. 9x 6y = 144 **Option 2: Both numbers in front of y are the SAME, but one is negative and one is positive. Either one of these options will work. 9x 8y = 24 Option 2: I can multiply the top by 4 Then I would have 4(3x + 2y = 48) (4)(3x) + (4)(2y) = (4)(48) 12x + 8y = 192 Now, compared with the second, the numbers infront of y are the same, but, one is negative. 12x + 8y = 192 9x 8y = 24 8

9 3. ADD the equations together. For this example I ll use Combine like terms. (One Option 2 from step 2. Variable should be eliminated). 12x + 8y = x 8y = 24 12x + 9x + 8y 8y = x = 168 (Combine like terms) 4. Solve this equation for the remaining variable. 21x = x 21 = x = 8 (Solve for x) 5. Plug this answer back into EITHER original equation 3x + 2y = 48 for x = 8, solve for y to solve for the remaining variable. OR 9x 8y = 24 for x = 8, solve for y I ll use the first one. 3(8) + 2y = y = y = y = Write your answer as an ordered pair. Answer: (8, 12) Check your work. 3(8) + 2(12) = 48 9(8) 8(12) = 24 9

10 Use the above steps to solve the following systems of equations by using addition. Example 1: Solve using addition (elimination). x + 3y = 5 2x 3y = 8 Example 2: Solve using addition (elimination). 7x = 5 2y 3y = 16 2x Example 3: Solve using addition (elimination). 4x + y = 10 6x y = 20 10

11 **Remember that when solving, you can also have either: INFINITE SOLUTIONS or NO SOLUTION. Example 4: Solve using addition (elimination). 4x + 6y = 12 6x + 9y = 12 What type of solution does this system have? This systems has Because. Example 5: Solve using addition (elimination). y 2x = 3 2y 4x = 6 What type of solution does this system have? This systems has Because. 11