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 of inconsistent.
`4x-5y =8 and 3x - (15)/(4) y=6`

Answer

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

1. \( 4x - 5y = 8 \) (Equation 1)
2. \( 3x - \frac{15}{4}y = 6 \) (Equation 2)

First, we will rewrite both equations in the standard form \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \).

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

For Equation 1:
\[ 4x - 5y - 8 = 0 \implies a_1 = 4, b_1 = -5, c_1 = -8 \]

For Equation 2:
\[ 3x - \frac{15}{4}y - 6 = 0 \implies a_2 = 3, b_2 = -\frac{15}{4}, c_2 = -6 \]

### Step 2: Calculate the ratios

Now, we will calculate the ratios \( \frac{a_1}{a_2}, \frac{b_1}{b_2}, \frac{c_1}{c_2} \).

1. Calculate \( \frac{a_1}{a_2} \):
\[ \frac{a_1}{a_2} = \frac{4}{3} \]

2. Calculate \( \frac{b_1}{b_2} \):
\[ \frac{b_1}{b_2} = \frac{-5}{-\frac{15}{4}} = \frac{-5 \times 4}{-15} = \frac{20}{15} = \frac{4}{3} \]

3. Calculate \( \frac{c_1}{c_2} \):
\[ \frac{c_1}{c_2} = \frac{-8}{-6} = \frac{8}{6} = \frac{4}{3} \]

### Step 3: Compare the ratios

Now we compare the ratios:
\[ \frac{a_1}{a_2} = \frac{4}{3}, \quad \frac{b_1}{b_2} = \frac{4}{3}, \quad \frac{c_1}{c_2} = \frac{4}{3} \]

Since:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = \frac{4}{3} \]

### Step 4: Conclusion

According to the rules of consistency:
- If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \), then the equations have infinitely many solutions and are consistent.

Thus, the given pair of linear equations is **consistent** and has **infinitely many solutions**.

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