To find the equation of a line parallel to a given line, we first need to understand the slope of the original line. The equation provided is:
37y = 89x + 7
We start by rewriting this equation in slope-intercept form (y = mx + b), where m represents the slope. To do this, we will isolate y on one side:
37y = 89x + 7
Now, divide each term by 37:
y = \frac{89}{37}x + \frac{7}{37}
From this, we can see that the slope m of the original line is:
m = \frac{89}{37}
Lines that are parallel have the same slope. Therefore, any line parallel to the original line will also have a slope of \frac{89}{37}. We can now use this slope to write the equation of the new line.
To write the equation of a parallel line, we need a point that the line will pass through. For our example, let’s say we want the new line to pass through the point (x1, y1). Using the point-slope form of the line’s equation:
y - y1 = m(x - x1)
This can be rewritten using our slope:
y - y1 = \frac{89}{37}(x - x1)
To create a specific equation, we need to choose coordinates (x1, y1). For instance, if we choose the point (1, 2), we plug in these values:
y - 2 = \frac{89}{37}(x - 1)
Expanding this, we get:
y - 2 = \frac{89}{37}x - \frac{89}{37}
Adding 2 to each side results in:
y = \frac{89}{37}x + \left(2 - \frac{89}{37}\right)
Calculating the constant gives:
y = \frac{89}{37}x + \frac{74 - 89}{37}
Which simplifies to:
y = \frac{89}{37}x - \frac{15}{37}
This is the equation of the line parallel to the given line that passes through the point (1, 2). In general, you will be able to substitute any point you desire for (x1, y1) to find the equivalent parallel line equation.