On comparing the ratios `(a_(1))/(a_(2)), (b_(1))/(b_(2)) and (c_(1))/(c_(2))`, find out whether the following pair of linear equations are consistent, or inconsistent.
`5x- 3y= 11, -10x + 6y= -22`

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 given equations are:
1. \( 5x - 3y = 11 \)
2. \( -10x + 6y = -22 \)

### Step 1: Rewrite the equations in standard form
We can rewrite both equations in the standard form \( ax + by + c = 0 \).

For the first equation:
\[ 5x - 3y - 11 = 0 \]

For the second equation:
\[ -10x + 6y + 22 = 0 \]

### Step 2: Identify coefficients
From the standard forms, we can identify the coefficients:
- For the first equation:
- \( a_1 = 5 \)
- \( b_1 = -3 \)
- \( c_1 = -11 \)

- For the second equation:
- \( a_2 = -10 \)
- \( b_2 = 6 \)
- \( c_2 = 22 \)

### Step 3: 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 \( \frac{a_1}{a_2} \):
\[ \frac{a_1}{a_2} = \frac{5}{-10} = -\frac{1}{2} \]

2. Calculate \( \frac{b_1}{b_2} \):
\[ \frac{b_1}{b_2} = \frac{-3}{6} = -\frac{1}{2} \]

3. Calculate \( \frac{c_1}{c_2} \):
\[ \frac{c_1}{c_2} = \frac{-11}{22} = -\frac{1}{2} \]

### Step 4: Compare the ratios
Now we compare the ratios:
- \( \frac{a_1}{a_2} = -\frac{1}{2} \)
- \( \frac{b_1}{b_2} = -\frac{1}{2} \)
- \( \frac{c_1}{c_2} = -\frac{1}{2} \)

Since all three ratios are equal, we have:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

### Conclusion
Since the ratios are equal, the pair of linear equations represents coincident lines, which means they have an infinite number of solutions. Therefore, the system of equations is consistent.