Chapter Systems of Equations

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1 SM-1 Name: Date: Hour: Chapter Systems of Equations 6.1- Solving Systems by Graphing CCS A.REI.6: Solve systems of equations exactly and approximately (e.g. with graphs), focusing on pairs of linear equations in two variables. Learning Goal: I will solve systems of equations by graphing. I will analyze special systems. Essential Understanding. You can use systems of linear equations to model problems. Systems of equations can be solved in more than one way. One method is to graph each equation and find the intersection point, if one exists. Vocabulary System of Linear Equations: Two or more linear equations. Solution of a System of Linear Equations: Any ordered pair that makes all of the equations in the system true. Consistent: A system of equations that has at least one solution. A consistent system can be either independent or dependent. A Consistent Independent System: A system of equations that has exactly one answer. A Consistent Dependent System: A system of equations that has infinitely many solutions. Inconsistent: A system of equations that has no solution.

2 Solving a System of Equations by Graphing: 1. What is the solution of the system? Use a graph. Y = x + 2 Y = 3x 2 Step 1. Graph each equation in the same coordinate plane. Step 2: Find the point of intersection. Check to see if that point makes each equation true. If yes, that is the solution of the system. Point of intersection? (, ) Check: Y = x + 2 y = 3x 2 Solution (, )

3 2. What is the solution of the system? Use a graph. Check your answer. Y = 2x + 4 Y = x + 2 Step 1: Graph each equation in the same coordinate plane. Step 2: Find the point of intersection. Check to see if that point makes each equation true. If yes, that is the solution of the system. Point of intersection? (, ) Check: y = 2x + 4 y = x + 2 Solution (, )

4 Writing a System of Equations 3. One satellite radio service charges $10 per month plus an activation fee of $20. A second service charges $11 per month plus an activation fee of $15. In what month was the cost of service the same? a. Write an equation for Service #1. b. Write an equation for Service #2. c. Graph both equations in the same coordinate plane. d. Point of intersection (, ) e. For what number of months is the cost the same? f. What is the cost for that many months?

5 Systems with Infinitely Many Solutions or No Solutions. 4. What is the solution of each system of equations? Use the graph to find your solution. (hint write the equations in slope-intercept form first, then graph) a. 3y = 3x 6 b. 2x = 3 y c. y = 1 2 x + 3 y = x 2 y = 2x 5 2y x = 1 Solution Solution Solution

6 5. Suppose one puppy weighs 5 pounds at birth and grows at a rate of 0.25 pounds per month over the next several months. Another puppy weighs 4 pounds at birth and grows at a rate of 0.5 pounds per month. After how many months will the puppies weigh the same amount? How much will they weigh? Puppy 1 Equation Puppy 2 Equation Point of intersection Explain the solution to the question in the context of the story Solving Systems Using Substitution CCS A.REI.6: Solve systems of equations exactly and approximately (e.g. with graphs), focusing on pairs of linear equations in two variables. Learning Goal: I will solve systems of equations using substitution. Essential Understanding. Systems of equations can be solved in more than one way. When a system has at least one equation that can be solved quickly for a variable, the system can be solved efficiently using substitution.

7 Using Substitution What is the solution of the system? Use substitution. y = 3x x + y = 32 Step 1: The first equation tells us that y = 3x, so substitute 3x for y in the second equation. y = 3x x + y = -32 to get x + 3x = -32 Step 2: Simplify and solve your new equation. x + 3x = -32 4x = -32 x = -8 Step 3: Put your answer for x back into both equations to find the y value. This is also your check. If both equations equal the same value, then your first answer is correct. Y = 3x x + y = -32 Y = 3(-8) -8 + y = -32 Y = y = The solution is (, )

8 6. Use substitution to find the solution to the system of equations. y = 2x + 7 y = x 1 Step 1: The first equation says that y is equal to 2x + 7 so you can substitute 2x + 7 in for y in the second equation. Y = 2x + 7 Y = x 1 Your new equation is Step 2: Solve your new equation. Step 3: Put your answer back into both equations to: 1) Find your second variable and 2) Check your first answer. Y = 2x + 7 y = x 1 Solution Solving for a Variable and Using Substitution 7. Use substitution to find the solution to this system of equations. 3y + 4x = 14-2x+ y = -3

9 Step 1: Solve one of the equations for one of the variables. Since the y in the second equation does not have a coefficient, that would be the easiest choice. Solve -2x + y = -3 for y Step 2: Substitute the expression for y into the y in the first equation and solve. 3( ) + 4x = 14 Step 3: Put your answer for x into both of the ORIGINAL equations and solve to check and to find your y value. 3y + 4x = 14-2x+ y = -3 Solution 8. Use substitution to solve the system of equations. 7x 3y = 2-2y + x = -6 a. Which equation would be easiest to use to solve for one variable? b. Solve for that variable c. Rewrite the other equation by substituting for that variable. d. Solve.

10 e. Check. f. Solution Using systems of Equations to solve problems. A snack bar sells 2 sizes of snack packs. A large snack pack is $5 and a small snack pack is $3. In one day, the snack bar sold 60 snack packs for a total of $220. How many small snack packs did the snack bar sell? Step 1: Write the system of equations. Call the $5 (big) snack packs x and the $3 (small) snack packs y. The first equation should represent the total number of snack packs sold. Translate the number of big packs sold, x, plus the number of small packs sold, y, is equal to 60 packs. The second equation should represent the money earned from selling the snack packs. Translate $5 times the number of big packs sold, plus $3 times the number of small packs sold is equal to $220. Step 2: Which equation would be easiest to use to solve for one variable? Solve for that variable Rewrite the other equation by substituting for that variable.

11 Solve. Check. Coordinate pair where these lines cross. Step 3: Make sure you answer the question asked. The question is asking for the number of snack packs sold. This was the variable in our equations, so the answer, in a complete sentence, is 9. The school bookstore sells t-shirts for $8 and sweatshirts for $12. Last month, the bookstore sold 37 t-shirts and sweatshirts for a total of $376. How many t- shirts were sold. Step 1: Write the system of equations. Define your variables: # of t-shirts sold # of sweatshirts sold Equation for number of shirts sold Equation for income from shirts sold Step 2: Solve an equation for 1 variable and substitute into the other equation. Solve.

12 Check. Coordinate pair where these lines cross Step 3: Answer the question asked. Systems with Infinitely Many Solutions or No Solution Remember: A system of equations can save exactly one solution, infinitely many solutions, or no solutions. If you get an identity, like 2 = 2, when you solve a system of equations, then the system has infinitely many solutions. If you get a false statement, like 8 = 2, then the system has no solution. 10. How many solutions does the system have? a) x = -3y + 4 6y + 2x = 8 b) y = 4x 9 y = 4x Printing a newsletter costs $1.50 per copy plus $450 in printer s fees. Copies are sold for $3 each. How many copies of the newsletter must be sold to break even? Solve using substitution. (the break-even point is were income = expenses) Equation for expenses: y = Equation for income: y =

13 Show your work to solve: Check: Solution in a complete sentence. 12. A group of scientists studied the effects of a chemical on various strains of bacteria. Strain A started with 6000 cells and decreased at a constant rate of 2000 cells per hour after the chemical was applied. Strain B started with 2000 cells and decreased at a constant rate of 1000 cells per hour after the chemical was applied. When will the strains have the same number of cells? Solve using substitution. Strain A: Strain B: Show your work to solve: Check: Solution in a complete sentence.

14 13. You have a jar of pennies and quarters. You want to choose 28 coins that are worth exactly $3.40. How many quarters do you have? Equation for number of coins: Equation for value of coins: Show your work to solve: Check: Solution in a complete sentence Solving Systems Using Elimination CCS A.REI.5: Prove that, given two equations, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Learning Goal: I will solve systems of equations by adding of subtracting to eliminate a variable. Essential Understanding. Systems of equations can be solved in more than one way. Some systems are written in a way that makes eliminating a variable a good method to use.

15 Solving a System by Adding Equations. What is the solution of the system. Use elimination. 2x + 5y = 17 6x 5y = -9 Step 1: Eliminate one variable. Notice that if you add the 2 equations together the y variables go away. 2x + 5y = x 5y = -9 8x + 0 = 8 Solve for x: 8x = 8 x = 1 Step 2: Substitute 1 in for x in each equation in order to find the value of y and to check. 2x + 5y = 17 6x 5y = -9 2(1) + 5y = 17 6(1) 5y = -9 (Continue) Solution 14. Use elimination to solve the system of equations. -3x 3y = 9 3x 4y = 5 Step 1: Eliminate one variable.

16 Step 2: Substitute the answer into each equation in order to find the value of the other variable and to check. Solution Solving a System by Subtracting Equations What is the solution of the system. Use elimination. x + y = x + y = 164 Step 1: Eliminate one variable. Notice that if you subtract the 2 equations together the y variables go away. x + y = (2.5 x + y = 164) -1.5x + 0 = -63 Solve for x: Step 2: Substitute the answer into each equation in order to find the value of the other variable and to check. Solution

17 15. Solve the system using elimination. (Remember that subtracting a negative is like ) 2x + 4y = 22 2x 2y = -8 Check: Solution Solving a System by Multiplying One Equation. Sometimes an extra step is needed before we can add or subtract to eliminate a variable. In the equations: -2x + 15y = -32 7x 5y = 17 The y variables would be eliminated if the 2 nd equation were multiplied by 3. -2x + 15y = -32-2x + 15y = -32 7x 5y = 17 multiply each term by x 15y = 51 Then add and solve.

18 Check: Put your answer for x into both of the ORIGINAL equations and solve to check and to find your y value. Solution 16. Use elimination to solve the system of equations: -5x 2y = 6 3x + 6y = 6 You can multiply the 1 st / 2 nd (circle one) equation by to change the equation in this system so we can solve it using elimination. Rewrite the system after multiplying: Solve by elimination: Check: Solution

19 Solving a System by Multiplying Both Equations. Sometimes you have to multiply BOTH equations by something to be able to use elimination to solve. 3x + 2y = 1 multiply by 3 9x + 6y = 3 4x + 3y = -2 multiply by 2 8x + 6y = -4 Then subtract and solve: (Be careful of subtracting negatives) Check: Use ORIGINAL equations! Solution 17. Solve using elimination. 4x + 3y = -19 multiply by 3x 2y = -10 multiply by Check: Use ORIGINAL equations! Add the equations and solve: Solution

20 18. A concession stand sold a total of 138 small and large popcorns. A small popcorn costs $2.50, and a large popcorn costs $4.00. Total popcorn sales were $ How many large popcorns were sold? Solve using elimination. Write the system of equations: Show your work to solve: Check: Solution in a complete sentence. 19. Farmer Fred has cows and chickens on his farm. His 525 animals have a total of 1600 legs. How many cows and chickens does he have on the farm? Solve using elimination. Write the system of equations: Show your work to solve: Check: Solution in a complete sentence.

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