Algebra 2: Systems of Equations Transition from Diagram to Algebra Part 1 (Complete in Class) Solve each of the following systems of equations. Try not to use diagrams, but to show the same logic using just algebra symbols. 1. x + 2y = 51 6x = 42 2. m + n = 11 m + 2n = 14 3. 2k + n = 11 6k + n = 19 4. 5u + v = 21 10u + 3v = 48 5. x + 4y = 5z x + 4y + 3z = 88 a. Solve for z only. b. What makes this problem “trickier” than the problems before it? c. Jenny claims that wherever you see the expression x + 4y, you can always replace it with 5z. Is she correct? Explain. 6. x + 4y = 5z x + 4y + 3z = 88 x + y = 31 (This is the same problem as #5 but now you need to solve for x and y as well.) 7. m = 2p 3m = 96 8. 4v + 4w = 56 v + 2w = 17 9. What makes problem #8 tricky (and what did you have to do to solve it)? 10. p + q + 0.7 = m m + p = 14 m + 4p = 29 11. h + p = -3k h + p + 7k = 104 Find k only. 12. If in problem #11, it is also given to you that h + 2p = -60, what is the value of p? Show work. 13. BIG IDEA: If an equation within a system contains multiple variables and you can’t solve it right away, how can you turn it into an equation that can be solved? Use the equations from problem #10 to help you be very specific in your explanation. 14. If I tell you that 3m + 12n = 15 within a system of equations, then what algebraic expression would equal 5? What algebraic expression would equal 10? Explain your answers clearly with either diagrams, algebra, or both! Part 2 (Complete for Homework): For each of the following problems, show step-by-step how you can solve it using algebra. You should not have to use diagrams! 1. 6x + 7y = 135 6x + 10y = 162 2. 5u + 6v = -4 5u + 6v + 10 = v 3. m + k = 6n m + k – 3n = 36 m = 2k 4. p = 7t p = 3t + 66 5. a + 3b = -3c a + 3b + 7c = 100 a + 4b = -70 6. p + r = 6 3p + q + 3r = 32 p + 2r = q 7. Your neighbors have three kids: James and Kimberly, who are twins, and Leslie, who is their older sister. The sum of the ages of all the kids is 38. The sum of just Leslie and Kimberly’s ages is 31. How old are they each? Here, write out the equation before answering the question. Try not to guess/check! Can you solve it systematically using the tools that we have been practicing?