To find the equation of the line that is parallel to $y = 3 - 5x$ and passes through the point $(1, 7)$, we need to follow these steps:

Step 1: Find the slope of the given line

The given equation is $y = 3 - 5x$, which is in the slope-intercept form $y = mx + b$, where:

• $m$ is the slope
• $b$ is the y-intercept

In this case, the slope $m$ of the line is $-5$ (since it’s the coefficient of $x$).

Step 2: Use the same slope for the new line

Since the lines are parallel, the new line will have the same slope, which is $-5$.

Step 3: Use the point-slope form of the equation

We will use the point-slope form of a line equation, which is:

$y - y_1 = m(x - x_1)$

where:

• $m = -5$ is the slope,
• $(x_1, y_1) = (1, 7)$ is the point through which the line passes.

Substitute these values into the equation:

$y - 7 = -5(x - 1)$

Step 4: Simplify to get the equation

Now, simplify the equation:

$y - 7 = -5x + 5$

Add 7 to both sides:

$y = -5x + 12$

Final Answer:

The equation of the line that is parallel to $y = 3 - 5x$ and passes through $(1, 7)$ is:

$y = -5x + 12$

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Would you like further explanation on any part of the solution? Here are some related questions:

1. How do you convert a line equation from point-slope form to slope-intercept form?
2. How can you find the slope between two points?
3. How do you write the equation of a line given two points?
4. What are some other forms of linear equations (besides point-slope)?
5. How do parallel and perpendicular lines relate to each other mathematically?

Tip: When working with parallel lines, remember that they have the same slope but different intercepts.