On comparing the ratios `(a_(1))/(a_(2)), (b_(1))/(b_(2)) and (c_(1))/(c_(2)) ,` find out whether the following pairs of linear equations are consistent or inconsistent.
`x - 5y =7 and -3x +15y =8`

Answer

To determine whether the given pair of linear equations is consistent or inconsistent, we will compare the ratios of the coefficients of the variables and the constant terms. The equations given are:

1. \( x - 5y = 7 \)
2. \( -3x + 15y = 8 \)

### Step 1: Rewrite the equations in standard form
We rewrite the equations in the form \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \).

1. The first equation can be rewritten as:
\[ x - 5y - 7 = 0 \quad \Rightarrow \quad 1x - 5y - 7 = 0 \]
Here, \( a_1 = 1 \), \( b_1 = -5 \), and \( c_1 = -7 \).

2. The second equation can be rewritten as:
\[ -3x + 15y - 8 = 0 \quad \Rightarrow \quad -3x + 15y - 8 = 0 \]
Here, \( a_2 = -3 \), \( b_2 = 15 \), and \( c_2 = -8 \).

### Step 2: Calculate the ratios
Now we will calculate the ratios \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), and \( \frac{c_1}{c_2} \).

1. Calculate \( \frac{a_1}{a_2} \):
\[ \frac{a_1}{a_2} = \frac{1}{-3} = -\frac{1}{3} \]

2. Calculate \( \frac{b_1}{b_2} \):
\[ \frac{b_1}{b_2} = \frac{-5}{15} = -\frac{1}{3} \]

3. Calculate \( \frac{c_1}{c_2} \):
\[ \frac{c_1}{c_2} = \frac{-7}{-8} = \frac{7}{8} \]

### Step 3: Compare the ratios
Now we compare the calculated ratios:

- \( \frac{a_1}{a_2} = -\frac{1}{3} \)
- \( \frac{b_1}{b_2} = -\frac{1}{3} \)
- \( \frac{c_1}{c_2} = \frac{7}{8} \)

### Step 4: Determine consistency
According to the conditions for consistency:
- The equations are consistent if \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) or \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).
- The equations are inconsistent if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) but \( \frac{c_1}{c_2} \neq \frac{a_1}{a_2} \).

In this case:
- \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) (both are \(-\frac{1}{3}\))
- \( \frac{c_1}{c_2} \neq \frac{a_1}{a_2} \) (since \(\frac{7}{8} \neq -\frac{1}{3}\))

Thus, the equations are inconsistent.

### Final Conclusion
The given pair of linear equations is **inconsistent**.

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