Find the equation using the points.
(1/2, -3/2) (-1/2, 1/2)
Answer: y=2x-1/2
Please explain why this is the answer.
If y = mx+b, plug in the points to get
1/2 m + b = -3/2
-1/2 m + b = 1/2
add them together to get
2b = -1
b = -1/2
subtract them to get
m = -2
y = mx+b = -2x - 1/2
or, consider the slope of the line:
(y2-y1)/(x2-x1) = (1/2+3/2)/(-1/2-1/2) = -2
Now use the point-slope form to get
y+3/2 = -2(x-1/2)
y = -2x+1-3/2 = -2x - 1/2
Better check that answer key
To find the equation using the given points, we can use the slope-intercept form of a linear equation, which is expressed as:
y = mx + b
where m is the slope of the line and b is the y-intercept.
To determine the slope (m), we can use the formula:
m = (y2 - y1) / (x2 - x1)
Given the points (1/2, -3/2) and (-1/2, 1/2), we can substitute these values into the formula to find the slope:
m = (1/2 - (-3/2)) / (-1/2 - 1/2)
= (1/2 + 3/2) / (-1/2 - 1/2)
= 4/2 / (-2/2)
= 4/2 / -2/2
= 4/-2
= -2
Now that we have the slope (m = -2), we can substitute this value along with one of the given points into the slope-intercept form:
y = mx + b
Using the point (1/2, -3/2):
-3/2 = -2(1/2) + b
Simplifying:
-3/2 = -1 + b
b = -3/2 + 2/2
b = -1/2
Therefore, the equation is:
y = -2x - 1/2
To find the equation of a line using two given points, you can use the slope-intercept form of the equation, y = mx + b, where m is the slope of the line and b is the y-intercept.
Step 1: Find the slope (m)
The slope (m) can be found using the formula:
m = (y2 - y1) / (x2 - x1)
Using the given points: (1/2, -3/2) and (-1/2, 1/2)
Let's denote the first point as (x1, y1) and the second point as (x2, y2):
x1 = 1/2, y1 = -3/2, x2 = -1/2, y2 = 1/2
Substituting the values into the formula:
m = (1/2 - (-3/2)) / (-1/2 - 1/2)
= (1/2 + 3/2) / (-1/2 - 1/2)
= (4/2) / (-2/2)
= 4 / -2
= -2
Therefore, the slope (m) is -2.
Step 2: Find the y-intercept (b)
To find the y-intercept, you can use either of the given points and substitute the values into the equation y = mx + b. Let's use the first point (1/2, -3/2).
Substituting the values into the equation:
-3/2 = -2 * (1/2) + b
-3/2 = -1 + b
b = -3/2 + 1
b = -3/2 + 2/2
b = -1/2
Therefore, the y-intercept (b) is -1/2.
Step 3: Write the equation
Now that we have both the slope (m = -2) and the y-intercept (b = -1/2), we can write the equation of the line in slope-intercept form:
y = mx + b
y = -2x - 1/2
Thus, the equation of the line passing through the given points is y = -2x - 1/2.