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If a, b and c are three different number...

If a, b and c are three different numbers and `ax : by : cz = 1 : 2 : -3,` then `ax + by +cz =`

A

`0`

B

`1//2`

C

`3`

D

`6`

Text Solution

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The correct Answer is:
To solve the problem, we need to find the value of \( ax + by + cz \) given the ratio \( ax : by : cz = 1 : 2 : -3 \). ### Step-by-Step Solution: 1. **Understanding the Ratio**: We are given that \( ax : by : cz = 1 : 2 : -3 \). This means that we can express \( ax \), \( by \), and \( cz \) in terms of a common variable. 2. **Setting Up the Variables**: Let's assume \( ax = k \), where \( k \) is a non-zero constant. According to the ratio: - \( by = 2k \) (since it corresponds to the second part of the ratio) - \( cz = -3k \) (since it corresponds to the third part of the ratio) 3. **Substituting the Values**: Now we can substitute these expressions into \( ax + by + cz \): \[ ax + by + cz = k + 2k + (-3k) \] 4. **Simplifying the Expression**: Combine the terms: \[ ax + by + cz = k + 2k - 3k = (1 + 2 - 3)k = 0k \] Thus, we have: \[ ax + by + cz = 0 \] 5. **Conclusion**: Therefore, the value of \( ax + by + cz \) is \( 0 \). ### Final Answer: \[ ax + by + cz = 0 \]
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