determine the equation of the line in y=mx+b form that passes through
1. (-8,1) & (-9, 2)
2. (3,7) & (-5, 9)
3, (-4,0) & (4,6)
Part B ) For each equation
Rewreite the y=mx+b equation in general form Ax+by+c=0 where A is positive b and C are integers
In each problem, put in x,y for each point set to get two equations to solve m,b
For example, in the first...
1=-8m+b and
2=-9m+b
solve these In this case subtract the bottom equation from the top
-1=m then solve for b.
k tnx
To determine the equation of a line in the form y = mx + b, we need to find the values of m and b. This can be done by using the given points and creating a system of equations.
Let's solve each problem step by step:
1. (-8,1) & (-9,2)
We can substitute the x and y values of the first point (-8,1) into the equation y = mx + b:
1 = -8m + b
Similarly, for the second point (-9,2):
2 = -9m + b
To solve this system of equations, we can subtract the second equation from the first equation:
(1) - (2) = (-8m + b) - (-9m + b)
-1 = m
Now we can substitute the value of m (-1) back into one of the original equations to find b. Let's use the first equation:
1 = -8m + b
1 = -8(-1) + b
1 = 8 + b
b = 1 - 8
b = -7
Therefore, the equation of the line passing through (-8,1) and (-9,2) is y = -x - 7.
For the second question, the procedure is the same:
2. (3,7) & (-5,9)
Substituting the first point (3,7):
7 = 3m + b
Substituting the second point (-5,9):
9 = -5m + b
Subtracting the second equation from the first one:
(7) - (9) = (3m + b) - (-5m + b)
-2 = 8m
Simplifying, we find:
m = -2/8
m = -1/4
Substituting m back into one of the original equations:
7 = 3m + b
7 = 3(-1/4) + b
7 = -3/4 + b
b = 7 + 3/4
b = 31/4
Therefore, the equation of the line passing through (3,7) and (-5,9) is y = (-1/4)x + 31/4.
Finally, for the third question:
3. (-4,0) & (4,6)
Substituting the first point (-4,0):
0 = -4m + b
Substituting the second point (4,6):
6 = 4m + b
Subtracting the second equation from the first equation:
(0) - (6) = (-4m + b) - (4m + b)
-6 = -8m
Simplifying, we find:
m = -6/-8
m = 3/4
Substituting m back into one of the original equations:
0 = -4m + b
0 = -4(3/4) + b
0 = -3 + b
b = 3
Therefore, the equation of the line passing through (-4,0) and (4,6) is y = (3/4)x + 3.
Now, for Part B:
To rewrite each equation in general form (Ax + By + C = 0), we need to multiply by a common factor to make sure A, B, and C are integers. Let's perform this conversion for each line:
1. y = -x - 7
Multiply every term by 1 to keep the equation unchanged:
1y + 1x - 7 = 0
A = 1, B = 1, C = -7
2. y = (-1/4)x + 31/4
Multiply every term by 4 to get rid of the fraction:
4y + (-1)x + 31 = 0
A = -1, B = 4, C = 31
3. y = (3/4)x + 3
Multiply every term by 4 to get rid of the fraction:
4y + 3x - 12 = 0
A = 3, B = 4, C = -12
Therefore, the general form of each equation is:
1. x + y - 7 = 0
2. -x + 4y + 31 = 0
3. 3x + 4y - 12 = 0