Write the equation of each line using the given information.
a. The points (−4, 1) and (2, 4) both lie on the line.
b. m = −1 and the point (2, −1) lies on the line.
c. It has the same slope as y = 5 and passes through (1, 1).
d. m = −3 and it has a y-intercept of (0, 8).
please help.
nvm i found the answers
a. To write the equation of the line using the given information, we need to first find the slope of the line, and then use one of the given points and the slope to construct the equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Step 1: Find the slope (m):
To find the slope (m), we use the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two given points. Let's use the points (-4, 1) and (2, 4) to find the slope:
m = (4 - 1) / (2 - (-4))
m = 3 / 6
m = 1/2
Step 2: Write the equation using slope-intercept form (y = mx + b):
We have the slope (m = 1/2), and we can choose either of the given points to find the y-intercept (b).
Let's choose the point (-4, 1) for convenience:
Using the point-slope form: (y - y1) = m(x - x1), we substitute the values:
(y - 1) = (1/2)(x - (-4))
y - 1 = (1/2)x + 2
y = (1/2)x + 3
Therefore, the equation of the line is y = (1/2)x + 3.
b. We are given that the slope (m) is -1, and a point (2, -1) lies on the line. To write the equation, we use the point-slope form (y - y1) = m(x - x1), where (x1, y1) is the given point and m is the slope:
Using the point-slope form with the given slope (-1) and point (2, -1):
(y - (-1)) = -1(x - 2)
y + 1 = -x + 2
y = -x + 2 - 1
y = -x + 1
Therefore, the equation of the line is y = -x + 1.
c. We are given that the line has the same slope as y = 5 and passes through the point (1, 1). Since the slope is already given, we can directly use it to construct the equation.
Using the slope-intercept form (y = mx + b), where m is the slope, we have:
y = 5x + b
To find the value of b (the y-intercept), we substitute the given point (1, 1) into the equation:
1 = 5(1) + b
1 = 5 + b
b = 1 - 5
b = -4
Therefore, the equation of the line is: y = 5x - 4.
d. We are given that the slope (m) is -3 and the line has a y-intercept of (0, 8). Using the slope-intercept form (y = mx + b), we can directly substitute the values into the equation:
Using the given slope (-3) and y-intercept (0, 8):
y = -3x + b
To find the value of b (the y-intercept), we substitute the given point (0, 8) into the equation:
8 = -3(0) + b
8 = 0 + b
b = 8
Therefore, the equation of the line is: y = -3x + 8.