Is the pair of equations `x - y =5 and 2y-x =10` inconsistent ? Justify your answer.

Answer

To determine if the pair of equations \( x - y = 5 \) and \( 2y - x = 10 \) is inconsistent, we will follow these steps:

### Step 1: Rewrite the equations in standard form
The given equations are:
1. \( x - y - 5 = 0 \)
2. \( -x + 2y - 10 = 0 \)

This can be rewritten as:
1. \( 1x + (-1)y + (-5) = 0 \) (Here, \( a_1 = 1, b_1 = -1, c_1 = -5 \))
2. \( -1x + 2y + (-10) = 0 \) (Here, \( a_2 = -1, b_2 = 2, c_2 = -10 \))

### Step 2: Calculate the ratios of coefficients
Using the coefficients from the equations, we will calculate:
- \( \frac{a_1}{a_2} \)
- \( \frac{b_1}{b_2} \)
- \( \frac{c_1}{c_2} \)

Calculating these:
- \( \frac{a_1}{a_2} = \frac{1}{-1} = -1 \)
- \( \frac{b_1}{b_2} = \frac{-1}{2} = -\frac{1}{2} \)
- \( \frac{c_1}{c_2} = \frac{-5}{-10} = \frac{1}{2} \)

### Step 3: Check the conditions for inconsistency
According to the condition for inconsistency:
- The equations are inconsistent if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) but \( \frac{c_1}{c_2} \) is not equal to either ratio.

Now, we check:
- \( \frac{a_1}{a_2} = -1 \)
- \( \frac{b_1}{b_2} = -\frac{1}{2} \)
- \( \frac{c_1}{c_2} = \frac{1}{2} \)

Since \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the condition for inconsistency is not satisfied.

### Conclusion
The pair of equations is not inconsistent, meaning they have at least one solution.