Consider the line 𝑦=2𝑥+2y=2x+2. If every point on the line is shifted 22 units up, find the equation of the new line.

1. ### Break Down the Problem

We need to find the equation of a line that results from shifting the original line $y = 2x + 2$ up by 22 units.

2. ### Relevant Concepts

Shifting a line upwards affects the y-intercept of the equation. If a line $y = mx + b$ is shifted up by $d$ units, the new line becomes: $y = mx + (b + d)$

3. ### Analysis and Detail

Here, the slope $m$ of the original line is 2, and the y-intercept $b$ is 2. We want to shift the line 22 units up:

• The new y-intercept will be: $b' = b + 22 = 2 + 22 = 24$

Thus, the equation of the new line will maintain the same slope (2) but have a new y-intercept (24).

4. ### Verify and Summarize

The new equation derived from the steps above is: $y = 2x + 24$

Final Answer

The equation of the new line after shifting up by 22 units is: $y = 2x + 24$