Final answer:
To find the required line parallel to 3x+7y=10 and passing through (-5,-9), determine the slope of the given line, which is -3/7. The new line will have the same slope, and using the point-slope form with the provided point gives us y = (-3/7)x - 78/7 as the final equation in slope-intercept form.
Step-by-step explanation:
To find the equation of a line that contains the point (-5,-9) and is parallel to the line with the equation 3x+7y=10, we first need to find the slope of the given line. The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. The slope of two parallel lines is the same, so we must first convert the given equation into slope-intercept form to identify the slope.
Starting with the given equation:
3x + 7y = 10
Subtract 3x from both sides to get:
7y = -3x + 10
Next, divide every term by 7 to solve for y:
y = (-3/7)x + 10/7
The slope (m) of the line is -3/7. Since the line we want to find must be parallel to this, its slope will also be -3/7.
Using the point (-5, -9) and the slope -3/7, we can plug these into the point-slope form of the equation of a line, which is:
y - y1 = m(x - x1), where (x1, y1) is the point the line passes through.
So, we have:
y - (-9) = (-3/7)(x - (-5))
y + 9 = (-3/7)(x + 5)
Now we distribute the slope:
y + 9 = (-3/7)x - (3/7)(5)
y + 9 = (-3/7)x - 15/7
Finally, we subtract 9 from both sides to get y by itself:
y = (-3/7)x - 15/7 - (9 * 7/7)
y = (-3/7)x - 63/7 - 15/7
y = (-3/7)x - 78/7
This is the equation of the line in slope-intercept form.