When writing an equation in point-slope format, is it not proper to fully solve for y?
For example, here is my problem:
Find the point-slope equation for a line containing the point (-7, 2) and the slope m = 3.
The answer was this:
y – 2 = 3(x + 7)
I was under the impression that you would solve for y by adding two to both sides, distributing the 3, etc, but a lot of answers I am seeing are showing like this. What is correct?
Thank you.
The equation y – 2 = 3(x + 7)
is considered point-slope form, since you can "see" the point and the slope.
solving for y, ...
y-2= 3x+21
y = 3x + 23 , gives you what is called the slope-yintercept form, because you can blatantly "see" the slope and the y-intercept
If you arrange it so that everything is on the left side, writing the x term first as a positive term, it is called the general form :
y - 2 = 3x + 21
-3x + y - 23 = 0
3x - y + 23 = 0
some texts and authors consider 3x - y = -23 to be "standard form"
What if both of your points were negative, such as (-7, -3), slope of 3
Would you write it as y - (-3) = 3(x - (-7)
and leave it like that, or would the correct answer in point-slope form be
y + 3 = 3(x + 7)
Thank you
When writing an equation in point-slope format, it is not necessary to fully solve for y. The point-slope form of a linear equation is y - y₁ = m(x - x₁), where (x₁, y₁) represents a point on the line and m represents the slope of the line.
In your example, you are given the point (-7, 2) and the slope m = 3. To find the point-slope equation, you can substitute these values into the formula:
y - 2 = 3(x + 7)
Here, you don't need to solve for y because the equation is already in point-slope form. The purpose of this form is to describe the relationship between the coordinates of a point on the line and its slope, rather than providing a fully solved equation.
However, if you really want to solve for y, you can do so by expanding the equation:
y - 2 = 3x + 21
By adding 2 to both sides, you get:
y = 3x + 23
This is called the slope-intercept form of the equation, where y is isolated on one side of the equation and the slope and y-intercept are clearly defined. But it is important to note that the point-slope form is still a valid representation of the equation of the line.
In summary, it is not necessary to fully solve for y when writing an equation in point-slope form. However, if you want to convert it to the slope-intercept form, you can solve for y by expanding and simplifying the equation.