Passes through the points (5,1) and (-1,4)
We can use the formula for the equation of a line given two points (x₁, y₁) and (x₂, y₂):
(y - y₁) = ((y₂ - y₁) / (x₂ - x₁)) * (x - x₁)
Using the given points (5,1) and (-1,4), we substitute the values into the formula:
(y - 1) = ((4 - 1) / (-1 - 5)) * (x - 5)
Simplifying,
(y - 1) = (-3/(-6)) * (x - 5)
(y - 1) = (1/2) * (x - 5)
Multiply both sides by 2 to get rid of the fraction:
2(y - 1) = x - 5
Distributing,
2y - 2 = x - 5
Rearranging the equation to the standard form,
x - 2y = -3
Therefore, the equation of the line that passes through the points (5,1) and (-1,4) is x - 2y = -3.
To pass through the points (5,1) and (-1,4), we can use the slope-intercept form of a linear equation, which is given by:
y = mx + b
where m is the slope of the line and b is the y-intercept.
First, let's find the slope (m). The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Let's substitute the given points into the formula:
m = (4 - 1) / (-1 - 5)
= 3 / -6
= -1/2
Now that we have the slope (m), we can substitute one of the given points and the slope into the slope-intercept form to find the y-intercept (b).
Using the point (5,1):
1 = (-1/2)(5) + b
1 = -5/2 + b
To find b, we can rearrange the equation:
b = 1 + 5/2
b = 2/2 + 5/2
b = 7/2
Now we have the slope (m) and the y-intercept (b). The equation of the line passing through the points (5,1) and (-1,4) is:
y = (-1/2)x + 7/2
To find the equation of a line that passes through two given points, we can use the point-slope form of a linear equation, which is given by:
y - y1 = m(x - x1),
where (x1, y1) are the coordinates of one of the points, m is the slope of the line, and (x, y) are the general coordinates on the line.
Given the points (5,1) and (-1,4), we can plug these values into the point-slope form equation to get the equation of the line.
Let's find the slope (m) first:
m = (y2 - y1) / (x2 - x1),
where (x1, y1) = (5,1) and (x2, y2) = (-1,4).
m = (4 - 1) / (-1 - 5) = 3 / -6 = -1/2.
Now, we can use any of the given points and the slope to substitute into the point-slope form equation. For simplicity, let's use the point (5,1):
y - 1 = -1/2(x - 5).
Expanding and simplifying this equation, we can find the final equation for the line that passes through the points (5,1) and (-1,4):
y - 1 = -1/2x + 5/2.
Re-arranging the equation, we can write it in standard form:
2y - 2 = -x + 5.
Or,
x + 2y = 7.
So, the equation of the line that passes through the points (5,1) and (-1,4) is x + 2y = 7.