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.
`(3)/(2) x+ (5)/(3) y= 7, 9x-10y =14`

Answer

To determine whether the given pair of linear equations is consistent or inconsistent, we will compare the ratios \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), and \( \frac{c_1}{c_2} \).

### Given equations:
1. \( \frac{3}{2} x + \frac{5}{3} y = 7 \)
2. \( 9x - 10y = 14 \)

### Step 1: Rewrite the equations in standard form
We can rewrite the first equation in standard form \( Ax + By + C = 0 \):
\[ \frac{3}{2} x + \frac{5}{3} y - 7 = 0 \]
To eliminate the fractions, we can multiply through by 6 (the least common multiple of 2 and 3):
\[ 6 \left( \frac{3}{2} x \right) + 6 \left( \frac{5}{3} y \right) - 6 \cdot 7 = 0 \]
This simplifies to:
\[ 9x + 10y - 42 = 0 \]

The second equation can also be rewritten:
\[ 9x - 10y - 14 = 0 \]

### Step 2: Identify coefficients
From the equations, we identify the coefficients:
- For the first equation \( 9x + 10y - 42 = 0 \):
- \( a_1 = 9 \)
- \( b_1 = 10 \)
- \( c_1 = -42 \)

- For the second equation \( 9x - 10y - 14 = 0 \):
- \( a_2 = 9 \)
- \( b_2 = -10 \)
- \( c_2 = -14 \)

### Step 3: Calculate the ratios
Now, we calculate the ratios:
1. \( \frac{a_1}{a_2} = \frac{9}{9} = 1 \)
2. \( \frac{b_1}{b_2} = \frac{10}{-10} = -1 \)
3. \( \frac{c_1}{c_2} = \frac{-42}{-14} = 3 \)

### Step 4: Compare the ratios
Now we compare the ratios:
- \( \frac{a_1}{a_2} = 1 \)
- \( \frac{b_1}{b_2} = -1 \)
- \( \frac{c_1}{c_2} = 3 \)

Since \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), the ratios are not equal.

### Conclusion
Since the ratios are not equal, the pair of linear equations is **inconsistent**. This means that the lines represented by these equations are parallel and will never intersect.

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