Question

This system of equations has an infinite number of solutions. Define the solutions algebraically, and allow z to represent all real numbers.

3x − 4y + 4z = 7
x − y − 2z = 2
2x − 3y + 6z = 5
x =

y =

z = all real numbers

Answer 1

x = 1+12z, y = -1+10z, and z = z

Explanation:

Step 1: Convert the system into the augmented matrix form:

• 3 -4 4 | 7

• 1 -1 -2 | 2

• 2 -3 6 | 5

Step 2: Multiply row 2 with -2 and add it into row 3:

• 3 -4 4 | 7

• 1 -1 -2 | 2

• 0 -1 10 | 1

Step 3: Multiply row 2 with -3 and add it into row 1:

• 0 -1 10 | 1

• 1 -1 -2 | 2

• 0 -1 10 | 1

Step 4: Replace row 1 with row 2 and multiply the updated row 2 with -1 and add it into row 3:

• 1 -1 -2 | 2

• 0 -1 10 | 1

• 0 0 0 | 0

Step 5: Multiply row 2 with -1 and add it in row 1:

0 1 -10 -1

• 1 0 -12 | 1

• 0 -1 10 | 1

• 0 0 0 | 0

Step 6: It can be seen that there are infinite solutions of this system since the last row is all zeroes. It can be seen that when this updated augmented matrix is converted into a system, it comes out to be:

• x - 12z = 1

• -y + 10z = 1

Step 7: Make x and y the subject of their respective equations:

• x = 1 + 12z

• y = -1 + 10z

So final answer is x = 1+12z, y = -1+10z, and z = z!!!