Find a consolidate ratio relation between x, y and z, if
`- 2x + 4y + 3z = 0`
` x - 3y + 5z = 0 `

Answer

To find a consolidated ratio relation between \( x \), \( y \), and \( z \) given the equations:

1. \(-2x + 4y + 3z = 0\)
2. \(x - 3y + 5z = 0\)

we can follow these steps:

### Step 1: Rewrite the equations in standard form
We have the two equations already in the standard form \( ax + by + cz = 0 \).

### Step 2: Identify coefficients
From the first equation, we identify:
- \( a_1 = -2 \)
- \( b_1 = 4 \)
- \( c_1 = 3 \)

From the second equation, we identify:
- \( a_2 = 1 \)
- \( b_2 = -3 \)
- \( c_2 = 5 \)

### Step 3: Set up the ratios
We can now set up the ratios of the coefficients:
\[ \frac{a_1}{a_2} = \frac{-2}{1}, \quad \frac{b_1}{b_2} = \frac{4}{-3}, \quad \frac{c_1}{c_2} = \frac{3}{5} \]

### Step 4: Check if the ratios are equal
Now we need to check if these ratios are equal:
- \(\frac{-2}{1} = -2\)
- \(\frac{4}{-3} = -\frac{4}{3}\)
- \(\frac{3}{5}\)

Since \(-2\) is not equal to \(-\frac{4}{3}\), the ratios are not equal.

### Step 5: Conclusion about the relationship
Since the ratios are not equal, it indicates that the two equations represent intersecting lines in three-dimensional space, which means there is a unique solution for \( x, y, z \).

### Step 6: Express the ratios
To express the consolidated ratio of \( x, y, z \), we can solve one of the equations for one variable in terms of the others. Let's solve the first equation for \( z \):

From \(-2x + 4y + 3z = 0\):
\[ 3z = 2x - 4y \implies z = \frac{2}{3}x - \frac{4}{3}y \]

Now we can express \( z \) in terms of \( x \) and \( y \).

### Step 7: Consolidated ratio
To find the consolidated ratio \( x:y:z \), we can express \( z \) in terms of \( x \) and \( y \):
Let \( x = 3k \) and \( y = k \) (for simplicity):
\[ z = \frac{2}{3}(3k) - \frac{4}{3}(k) = 2k - \frac{4}{3}k = \frac{6k - 4k}{3} = \frac{2k}{3} \]

Thus, the consolidated ratio \( x:y:z \) can be expressed as:
\[ x:y:z = 3k:k:\frac{2k}{3} = 9:3:2 \]

### Final Answer
The consolidated ratio relation between \( x, y, z \) is:
\[ x:y:z = 9:3:2 \]