A line has a slope of 6 and an x-intercept of 7.
a. Write the equation for the line in slope-intercept form. Justify your work. The equation for the line in slope-intercept form is y = 6x + 7. This can be justified by using the slope-intercept form of a line, which is y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is 6 and the x-intercept is 7, so the equation is y = 6x + 7.
b. Another line, with the same slope as the first, passes through the point (-1, -1). Is enough information provided to write the equation of this line? Explain. Find the equation if one can be written.
Not quite!
First of all in the original, which is not a question but merely a statement,
it said:
for y = 6x + 7
"the slope is 6 and the x-intercept is 7, so the equation is y = 6x + 7."
That should be the y-intercept is 7, not the x-intercept
secondly, a line parallel to y = 6x + 7 and passing through (-1,-1)
should only differ in the constant
so we start with y = 6x + b , but (-1,-1) lies on it, so
-1 = 6(-1) + b
b = 5
y = 6x + 5
or , the way the bot did it
y+1 = 6(x+1)
y = 6x + 6 - 1
y = 6x + 5
In order to write the equation of a line, we need both the slope and a point on the line. In this case, the same slope of 6 is provided, but we are also given a point (-1, -1). With this additional information, we can now write the equation of the line.
To find the equation, we can use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. Plugging in the values (-1, -1) for (x1, y1) and 6 for m, we get:
y - (-1) = 6(x - (-1))
y + 1 = 6(x + 1)
Simplifying further:
y + 1 = 6x + 6
Finally, we can write the equation in slope-intercept form by isolating y:
y = 6x + 6 - 1
y = 6x + 5
Therefore, the equation of the line passing through the point (-1, -1) with the slope of 6 is y = 6x + 5.
To determine if enough information is provided to write the equation of the second line, we need to check if we have the slope and a point on the line. In this case, we are given the slope (6) but only have one point (-1, -1).
To find the equation of the second line, we can use the point-slope form of a line equation, which is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope.
Plugging in the values we have, we get:
y - (-1) = 6(x - (-1))
y + 1 = 6(x + 1)
Now, we can simplify this equation:
y + 1 = 6x + 6
To write the equation in slope-intercept form, y = mx + b, we can rearrange the equation:
y = 6x + 6 - 1
y = 6x + 5
Therefore, the equation of the second line with the same slope as the first and passing through the point (-1, -1) is y = 6x + 5.