In Exercises $$ 37-40 $$, find three solutions to the system of equations, 37. $$ \left\{\begin{array}{r}x+2 y+z=5 \ y+3 z=9\end{array} ight. $$ 38. $$ \left\{\begin{array}{r}x+5 y+3 z=9 \ 2 x+9 y+7 z=5\end{array} ight. $$ 39. $$ \left\{\begin{array}{r}x+7 y-3 z=8 \ z=5\end{array} ight. $$ 40. $$ \left\{\begin{array}{r}x \ y-3 z=7\end{array} ight. $$

Question

In Exercises $$ 37-40 $$, find three solutions to the system of equations, 37. $$ \left\{\begin{array}{r}x+2 y+z=5 \ y+3 z=9\end{array}ight. $$ 38. $$ \left\{\begin{array}{r}x+5 y+3 z=9 \ 2 x+9 y+7 z=5\end{array}ight. $$ 39. $$ \left\{\begin{array}{r}x+7 y-3 z=8 \ z=5\end{array}ight. $$ 40. $$ \left\{\begin{array}{r}x \ y-3 z=7\end{array}ight. $$

UpStudy AI · Accepted Answer

We are given systems with two equations in three variables. In each case there will be one free parameter, and in the last problem only one equation appears, so two variables are free. Your task is to choose three specific solutions that satisfy the given equations.

Let’s go through each exercise.

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Exercise 37:
  The system is
    x + 2y + z = 5    (1)
    y + 3z = 9      (2)

Step 1. Solve (2) for y:
  y = 9 − 3z.

Step 2. Substitute y in (1):
  x + 2(9 − 3z) + z = 5  
  x + 18 − 6z + z = 5  
  x − 5z = 5 − 18  
  x − 5z = −13  
  x = 5z − 13.

Let z be a free parameter (say t). Then the general solution is:
  x = 5t − 13,
  y = 9 − 3t,
  z = t.

Choose three values for t, for example:
 • t = 0: (x, y, z) = (−13, 9, 0)
 • t = 1: (x, y, z) = (−8, 6, 1)
 • t = 2: (x, y, z) = (−3, 3, 2)

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Exercise 38:
  The system is
    x + 5y + 3z = 9    (1)
    2x + 9y + 7z = 5    (2)

Step 1. Solve (1) for x:
  x = 9 − 5y − 3z.

Step 2. Substitute x into (2):
  2(9 − 5y − 3z) + 9y + 7z = 5  
  18 − 10y − 6z + 9y + 7z = 5  
  18 − y + z = 5  
  −y + z = 5 − 18  
  −y + z = −13  ⟹ y − z = 13  
  Therefore, y = z + 13.

Step 3. Substitute y back into the expression for x:
  x = 9 − 5(z + 13) − 3z  
  x = 9 − 5z − 65 − 3z  
  x = −56 − 8z.

Now, with the free parameter t in place of z, the general solution is:
  x = −56 − 8t,
  y = t + 13,
  z = t.

Choose three values for t, for example:
 • t = 0: (x, y, z) = (−56, 13, 0)
 • t = 1: (x, y, z) = (−64, 14, 1)
 • t = −1: (x, y, z) = (−48, 12, −1)

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Exercise 39:
  The system is
    x + 7y − 3z = 8    (1)
    z = 5        (2)

Step 1. Substitute z = 5 into (1):
  x + 7y − 3(5) = 8  
  x + 7y − 15 = 8  
  x + 7y = 23.

Let y be the free parameter (say t). Then solve for x:
  x = 23 − 7t.

Thus, the general solution is:
  x = 23 − 7t,
  y = t,
  z = 5.

Choose three values for t, for example:
 • t = 0: (x, y, z) = (23, 0, 5)
 • t = 1: (x, y, z) = (16, 1, 5)
 • t = 2: (x, y, z) = (9, 2, 5)

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Exercise 40:
  The system is written as
    x
    y − 3z = 7

This system lists only one equation involving y and z and leaves x “free” (there is no condition on x). Thus, x can be any real number, and the equation y − 3z = 7 must hold.

Step 1. Solve for y in terms of z:
  y = 7 + 3z.

Let x and z be free parameters. A general solution for (x, y, z) is:
  x = s  (where s is any real number),
  y = 7 + 3t  (where t is any real number),
  z = t.

Choose specific values. For example, we may select:
 • For s = 0 and t = 0: (x, y, z) = (0, 7, 0)
 • For s = 1 and t = 1: (x, y, z) = (1, 10, 1)
 • For s = −2 and t = −1: (x, y, z) = (−2, 4, −1)

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Summary of Answers:

37. General solution: x = 5t − 13, y = 9 − 3t, z = t.
  Three examples: (−13, 9, 0), (−8, 6, 1), (−3, 3, 2).

38. General solution: x = −56 − 8t, y = 13 + t, z = t.
  Three examples: (−56, 13, 0), (−64, 14, 1), (−48, 12, −1).

39. General solution: x = 23 − 7t, y = t, z = 5.
  Three examples: (23, 0, 5), (16, 1, 5), (9, 2, 5).

40. General solution: x = s, y = 7 + 3t, z = t.
  Three examples: (0, 7, 0), (1, 10, 1), (−2, 4, −1).

Each set of three solutions satisfies the corresponding system.
Here are three solutions for each of the four systems: **Exercise 37:** 1. (−13, 9, 0) 2. (−8, 6, 1) 3. (−3, 3, 2) **Exercise 38:** 1. (−56, 13, 0) 2. (−64, 14, 1) 3. (−48, 12, −1) **Exercise 39:** 1. (23, 0, 5) 2. (16, 1, 5) 3. (9, 2, 5) **Exercise 40:** 1. (0, 7, 0) 2. (1, 10, 1) 3. (−2, 4, −1) Each set satisfies the respective system of equations.

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