Are the following pair of linear equations consistent? Justify your answer.
`-3x-4y= 12, 4y+ 3x= 12`

Answer

To determine whether the given pair of linear equations is consistent, we can follow these steps:

### Given Equations:
1. \(-3x - 4y = 12\) (Equation 1)
2. \(4y + 3x = 12\) (Equation 2)

### Step 1: Rewrite the equations in standard form
The standard form of a linear equation is \(Ax + By = C\).

For Equation 2, we can rearrange it:
\[ 3x + 4y = 12 \]

Now we have the two equations:
1. \(-3x - 4y = 12\)
2. \(3x + 4y = 12\)

### Step 2: Identify coefficients
From the equations, we can identify the coefficients:
- For Equation 1: \(a_1 = -3\), \(b_1 = -4\), \(c_1 = 12\)
- For Equation 2: \(a_2 = 3\), \(b_2 = 4\), \(c_2 = 12\)

### Step 3: Calculate the ratios
Now, we will calculate the ratios \(\frac{a_1}{a_2}\) and \(\frac{b_1}{b_2}\):
\[ \frac{a_1}{a_2} = \frac{-3}{3} = -1 \]
\[ \frac{b_1}{b_2} = \frac{-4}{4} = -1 \]

### Step 4: Check the condition for consistency
For the system of equations to be consistent, we need to check if:
\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]
In this case:
\[ -1 = -1 \]
Since both ratios are equal, the system of equations is inconsistent.

### Conclusion
The given pair of linear equations is inconsistent because the ratios of the coefficients of \(x\) and \(y\) are equal.

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