Q 4.69 - Intermediate Algebra

Question

Solve the system by equations:

$$\begin{gathered} x + y - z = 0 \\ 2x + 4y - 2z = 6 \\ 3x + 6y - 3z = 9 \end{gathered}$$

Step-by-Step Solution

Answer

There are infinitely many number of solutions of given system of equations.

Step 1. Given Information

We are given a system of linear equation, 

$$\begin{gathered} x + y - z = 0 -\left(1\right) \\ 2x + 4y - 2z = 6 -\left(2\right) \\ 3x + 6y - 3z = 9 -\left(3\right) \end{gathered}$$

Step 2. Solving the equations

Multiplying by $3$ in second equation and $2$ in third equation, we get

$$\begin{gathered} 3(2x+4y-2z)=3\left(6\right) \\ 6x+12y-6z=18 -\left(4\right) \end{gathered}$$

$$\begin{gathered} 2(3x+6y-3z=2(9) \\ 6x+12y-6z=18 -(5) \end{gathered}$$

Now, subtracting the fourth and fifth equations, we get

$$\begin{gathered} 6x-6x+12y-12y-6z+6z=18-18 \\ 0=0 \end{gathered}$$

This is true, hence the system has infinitely many solutions.