What is the equation in point-slope form of the line passing through (2, 5) and (−1, 8)?
Point slope form is just y - y₁ = m (x - x₁)
so now the task facing you is to find the slope. That is, of course
m = (y2-y1)/(x2-x1) = 3/-3 = -1
So now you can use point 1 to get
y-5 = -1(x-2)
or you can use point 2, since the equation holds for ANY point on the line.
y-8 = -1(x+1)
It is easy to see that these are both the same line
To find the equation of a line in point-slope form, use the formula:
\[y - y_1 = m(x - x_1)\]
where (x1, y1) is a point on the line, and m is the slope of the line.
Step 1: Find the slope (m)
The slope (m) can be calculated using the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
where (x1, y1) and (x2, y2) are two points on the line.
Given points:
Point 1: (2, 5)
Point 2: (-1, 8)
Using the formula:
\[m = \frac{8 - 5}{-1 - 2}\]
m = \(\frac{3}{-3}\) = -1
Step 2: Substitute the slope and one point into the point-slope formula
Using the point-slope formula, we can choose either point to substitute. Let's use point 1 (2, 5).
Substituting the values into the formula:
\[y - 5 = -1(x - 2)\]
Simplifying:
\[y - 5 = -x + 2\]
Step 3: Rewrite the equation in point-slope form
To rewrite the equation in point-slope form, we can rearrange it:
\[y = -x + 7\]
So, the equation in point-slope form of the line passing through (2, 5) and (-1, 8) is \(y = -x + 7\).
To find the equation of a line in point-slope form, we need to know the coordinates of a point on the line and the slope of the line.
Step 1: Calculate the slope of the line.
The slope (m) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Let's denote the coordinates of the two given points as:
Point 1: (x1, y1) = (2, 5)
Point 2: (x2, y2) = (-1, 8)
Using the formula to calculate the slope, we have:
m = (8 - 5) / (-1 - 2)
= (3) / (-3)
= -1
Step 2: Use the point-slope form to write the equation.
The equation of a line in point-slope form is given as:
y - y1 = m(x - x1)
Substituting the known values:
(x1, y1) = (2, 5) and m = -1
The equation becomes:
y - 5 = -1(x - 2)
Simplifying further:
y - 5 = -x + 2
y = -x + 7
Therefore, the equation in point-slope form of the line passing through (2, 5) and (-1, 8) is y = -x + 7.