Write an equation in slope-intercept form, point-slope, or standard form for the line with the given information. Explain why you chose the form you used. a. Passes through (-1, 4) and (-5, 2)
To find the equation of the line passing through the points (-1, 4) and (-5, 2), we can use either the point-slope form or the slope-intercept form.
1. Point-Slope Form:
The point-slope form of a linear equation is given by: y - y₁ = m(x - x₁), where (x₁, y₁) represents a point on the line and m is the slope of the line.
Let's first calculate the slope using the two given points:
m = (y₂ - y₁) / (x₂ - x₁)
= (2 - 4) / (-5 - (-1))
= -2 / (-4)
= 1/2
Now, substituting one of the given points (-1, 4) into the point-slope form, we get:
y - 4 = (1/2)(x - (-1))
Simplifying the equation:
y - 4 = (1/2)(x + 1)
y - 4 = (1/2)x + 1/2
So, the equation of the line in point-slope form is y - 4 = (1/2)x + 1/2.
2. Slope-Intercept Form:
The slope-intercept form of a linear equation is given by: y = mx + b, where m is the slope of the line and b is the y-intercept (the y-coordinate where the line crosses the y-axis).
To find the equation in slope-intercept form, we need to determine the slope and the y-intercept.
Using the slope formula, we have already determined the slope to be 1/2.
Now, we can substitute one of the given points, for example, (-1, 4), and solve for the y-intercept b:
4 = (1/2)(-1) + b
4 = -1/2 + b
4 + 1/2 = b
b = 9/2
So, the slope-intercept form of the equation is y = (1/2)x + 9/2.
In summary, the equation of the line that passes through (-1, 4) and (-5, 2) can be written in either point-slope form as y - 4 = (1/2)(x + 1) or in slope-intercept form as y = (1/2)x + 9/2.
To write the equation of a line, we need either the slope and y-intercept (slope-intercept form), the slope and a point (point-slope form), or two points (standard form).
To determine the slope of the line passing through (-1, 4) and (-5, 2), we can use the slope formula:
slope = (y2 - y1) / (x2 - x1)
Substituting the given points, we have:
slope = (2 - 4) / (-5 - (-1))
= -2 / (-5 + 1)
= -2 / (-4)
= 1/2
Now that we have the slope, we can proceed with choosing a suitable form to write the equation.
Option 1: Slope-Intercept Form (y = mx + b)
In this form, "m" represents the slope and "b" represents the y-intercept. However, we do not have the y-intercept of the line, so we cannot use this form.
Option 2: Point-Slope Form (y - y1 = m(x - x1))
In this form, "m" represents the slope, and (x1, y1) represents a point on the line. We have the slope (1/2) and two points (-1, 4) and (-5, 2). We can choose either point to substitute into the formula. For simplicity, let's choose (-1, 4):
y - 4 = (1/2)(x - (-1))
y - 4 = (1/2)(x + 1)
y - 4 = (1/2)x + 1/2
y = (1/2)x + 1/2 + 4
y = (1/2)x + 9/2
So, the equation of the line passing through (-1, 4) and (-5, 2) can be written in point-slope form as y = (1/2)x + 9/2.
Option 3: Standard Form (Ax + By = C)
In this form, "A," "B," and "C" represent coefficients. To use this form, we need to manipulate the equation we obtained in point-slope form to isolate "x" and "y":
y = (1/2)x + 9/2
Multiply both sides by 2 to eliminate the fraction:
2y = x + 9
Now, let's rearrange the equation to fit the standard form:
x - 2y = -9
So, the equation of the line passing through (-1, 4) and (-5, 2) in standard form is x - 2y = -9.
To summarize, we chose point-slope form because we had the slope and one point given. Then, we transformed the equation into standard form to complete our answer.
What is your question on this assignment?
I would use point slope, but it could be any of the forms.
(-1,4) (-5,2)
Slope: m = y2 - y1/x2 - x1
2 - 4 / -5 - (-1) = -2/-4
m = 1/2
Equation: y - y1 = m(x - x1)
y - 4 = 1/2(x - (-1))
y - 4 = 1/2(x + 1)
y - 4 + 4 = 1/2x + 1/2 + 4
y = 1/2x + 9/2