Every point on a particular line is shifted 66 units down. The resulting line has equation𝑦=2𝑥−2y=2x−2. Find the equation of the original line.

1. Break Down the Problem

To find the equation of the original line, we start with the given equation of the shifted line, which is $y = 2x - 2$. Since each point on the original line is shifted 66 units down, we can say that if the original line has an equation $y = mx + b$, then shifting it down will result in the new equation:

$y - 66 = mx + b$

2. Relevant Concepts

When we describe a vertical shift in the coordinate plane:

• Shifting down means that we decrease the $y$-value of every point by 66 units. Therefore, the equation of the original line can be expressed as:

$y = mx + b + 66$

Given the new line equation $y = 2x - 2$, we can equate this with our shift equation:

3. Analysis and Detail

Setting the equations equal:

$mx + b + 66 = 2x - 2$

From here, we’ll equate the slopes and the intercepts to find $m$ and $b$.

1. Equating the slopes: $m = 2$

2. Substituting $m$ into the intercept equation: $b + 66 = -2$

3. Solving for $b$: $b = -2 - 66 = -68$

4. Verify and Summarize

Now, substituting $m$ and $b$ back into the equation for the original line:

$y = 2x - 68$

This is the equation of the original line. To verify, we can check if shifting this equation down by 66 gives $y = 2x - 2$:

$y + 66 = 2x - 68 + 66 \Rightarrow y = 2x - 2$

Both equations match, confirming our solution.

Final Answer

The equation of the original line is $y = 2x - 68$.