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.
`(4)/(3)x + 2y= 8, 2x + 3y= 12`

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. \( \frac{4}{3}x + 2y = 8 \)
2. \( 2x + 3y = 12 \)

### Step 1: Rewrite the equations in standard form

The standard form of a linear equation is \( ax + by + c = 0 \).

For the first equation:
\[ \frac{4}{3}x + 2y - 8 = 0 \]
This can be rewritten as:
\[ \frac{4}{3}x + 2y + (-8) = 0 \]

For the second equation:
\[ 2x + 3y - 12 = 0 \]

### Step 2: Identify coefficients

From the first equation \( \frac{4}{3}x + 2y - 8 = 0 \):
- \( a_1 = \frac{4}{3} \)
- \( b_1 = 2 \)
- \( c_1 = -8 \)

From the second equation \( 2x + 3y - 12 = 0 \):
- \( a_2 = 2 \)
- \( b_2 = 3 \)
- \( c_2 = -12 \)

### 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 \( \frac{a_1}{a_2} \):
\[ \frac{a_1}{a_2} = \frac{\frac{4}{3}}{2} = \frac{4}{3} \times \frac{1}{2} = \frac{4}{6} = \frac{2}{3} \]

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

3. Calculate \( \frac{c_1}{c_2} \):
\[ \frac{c_1}{c_2} = \frac{-8}{-12} = \frac{8}{12} = \frac{2}{3} \]

### Step 4: Compare the ratios

Now we compare the ratios:
- \( \frac{a_1}{a_2} = \frac{2}{3} \)
- \( \frac{b_1}{b_2} = \frac{2}{3} \)
- \( \frac{c_1}{c_2} = \frac{2}{3} \)

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

### Conclusion

When \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \), the lines represented by the equations are coincident, which means they are consistent and have infinitely many solutions.

Thus, the pair of linear equations is **consistent**.

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