What will be the new equation of straight line 3x + 4y = 6 if the origin gets shifted to (3, - 4)?

Options

A. 3 x + 4y = 5
B. 4x - 3y = 4
C. 3 x + 4y + 1 = 0
D. 3x + 4y - 13 =0

Answer

To find the new equation of the straight line \(3x + 4y = 6\) after shifting the origin to the point \((3, -4)\), we can follow these steps:

### Step 1: Understand the Shift in Coordinates
When the origin shifts from \((0, 0)\) to \((3, -4)\), we need to express the new coordinates \(X\) and \(Y\) in terms of the old coordinates \(x\) and \(y\).

The transformations are:
- \(X = x + 3\) (since we are moving the x-coordinate by +3)
- \(Y = y - 4\) (since we are moving the y-coordinate by -4)

### Step 2: Substitute the New Coordinates into the Original Equation
We start with the original equation:
\[ 3x + 4y = 6 \]

Now, we will replace \(x\) and \(y\) with their expressions in terms of \(X\) and \(Y\):
- From \(X = x + 3\), we get \(x = X - 3\)
- From \(Y = y - 4\), we get \(y = Y + 4\)

Substituting these into the original equation:
\[ 3(X - 3) + 4(Y + 4) = 6 \]

### Step 3: Simplify the Equation
Now, we simplify the equation:
\[ 3X - 9 + 4Y + 16 = 6 \]
Combine like terms:
\[ 3X + 4Y + 7 = 6 \]

### Step 4: Rearrange the Equation
Now, we rearrange the equation to standard form:
\[ 3X + 4Y + 7 - 6 = 0 \]
This simplifies to:
\[ 3X + 4Y + 1 = 0 \]

### Final Result
Thus, the new equation of the straight line after shifting the origin to \((3, -4)\) is:
\[ 3X + 4Y + 1 = 0 \]

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