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How Do You Solve a Word Problem Using Two Equations?

Alice and Bill like to save their weekly allowances to buy algebra supplies. One time Alice saved for one week, Bill saved for two weeks and they bought an algebra book for $23. Another time Alice and Bill each saved for 3 weeks and bought matching gold 'I love algebra' necklaces for $48 total. If they never had any money left over what are each of their weekly allowances?

Summary

  1. Write equations based on the problem
  2. Solve a+2b=23 for a to get a=23-2b
  3. Plug 23-2b in for 'a' in 3a+3b=48
  4. Solving for b, we get b=7
  5. Plug b=7 into a+2b=23 to find a
  6. Solving for a, we get a=9

Notes

    1. 'a' is a variable representing Alice's weekly allowance
    1. 'b' is a variable representing Bill's weekly allowance
    1. 'a' is a variable representing Alice's weekly allowance
    1. 'b' is a variable representing Bill's weekly allowance
    2. Since 'b' is his allowance for one week, we multiply by 2 to get his allowance for two weeks
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. Combined, they are 'a + 2b'
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. That sum is 'a + 2b' and we set that equal to 23
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. We multiply by 3 to get their allowance for three weeks, 3(a + b) = 3a + 3b
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. Combined, they are '3a + 3b'
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. The left side is '3a + 3b' and we set that equal to 48
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. The first equation is 'a+3b=23'
    4. The second equation is '3a+3b=48'
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    1. Working with the first equation, subtract '2b' from both sides
    2. 'a' is a variable representing Alice's weekly allowance
    3. 'b' is a variable representing Bill's weekly allowance
    1. 'b' is a variable representing Bill's weekly allowance
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. Plug '23-2b' into the second equation for every place you see an 'a' variable
    1. 'b' is a variable representing Bill's weekly allowance
    2. 3 • 23 = 69
    3. 3 • -2b = -6b
    4. After we distribute, '3 • (23 - 2b)' becomes '69 - 6b'
    1. 'b' is a variable representing Bill's weekly allowance
    2. '-6b' plus '3b' is equal to '-3b'
    1. 'b' is a variable representing Bill's weekly allowance
    1. 'b' is a variable representing Bill's weekly allowance
    2. 69 - 3b - 69 = -3b
    3. 48 - 69 = 21
    1. 'b' is a variable representing Bill's weekly allowance
    2. '-3b' divided by '-3' is just 'b'
    3. '-21' divided by '-3' is equal to '7'
    1. 'b' is a variable representing Bill's weekly allowance
    2. b = 7
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. The first equation becomes:
    4. a + 2 • 7 = 23
    5. 2 • 7 = 14
    1. 'a' is a variable representing Alice's weekly allowance
    2. a + 14 - 14 = a
    3. 23 - 14 = 9
    1. 'a' is a variable representing Alice's weekly allowance
    2. a = 9
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. Let's check to see if 'a=9' and 'b=7' are correct answers
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. Remember, 'a=9' and 'b=7' are being plugged into our first equation
    4. Simplify the equation:
    5. 2 • 7 = 14
    6. 9 + 14 = 23
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. Our first equation is true when 'a=9' and 'b=7'
    4. Remember, both equations have to be true when 'a=9' and 'b=7' for our answers to be correct
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. Now 'a=9' and 'b=7' are being plugged into our second equation
    4. Simplify the equation:
    5. 3 • 9 = 27
    6. 3 • 7 = 21
    7. 27 + 21 = 48
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. Our second equation is true when 'a=9' and 'b=7'
    1. 'a' is a variable representing Alice's weekly allowance
    2. 'b' is a variable representing Bill's weekly allowance
    3. Both equations are to be true when 'a=9' and 'b=7'!
    4. Our answers must be correct!