Hello,
Slope intercept form is y = mx + b, where m is the slope and b is the constant.
1. The slope is already given to us, so we know m = -3. We know the y and x value of one point of the line, so we can plug it into the equation to solve for b.
So, we get:
6 = -3(1) + b
6 = b - 3
b = 9
So, the equation for number 1 is y = -3x + 9.
To make sure, check the answer by checking if the equation passes through (1,6).
6 = -3(1) + 9
6 = 9 - 3
6 = 6, checks out.
The line also has a slope of -3, represented by m.
y = -3x + 9
2. This is a similar problem just with different numbers, so I'll make the process a little shorter this time.
m = 1/2, as given in the question.
y = mx + b
y = (1/2)x + b
7 = (1/2)12 + b
7 = 6 + b
b = 1
So, we have the equation y = (1/2)x + 1.
Check the answer again with the question, just to make sure.
y = (1/2)x + 1
7 = (1/2)12 + 1
7 = 6 + 1
7 = 7, checks out.
y = (1/2)x + 1
3. This time we are given two points. To find the slope of a line given two points on the line, the formula is (y2-y1) / (x2-x1). In this situation, x1 = 1, y1 = -7, x2 = 5, and y2 = 1.
So, the slope is (y2-y1) / (x2 - x1) = (1 - (-7)) / (5 - 1) = (8) / (4) = 2.
So, m = 2
We then plug n what we know about the equation of the line to find b in the equation y = mx + b.
So, we have y = mx + b
-7 = 2(1) + b
-7 = b + 2
b = -7 - 2 = -9
This gives us the equation, y = 2x - 9.
Check it with the other point too, just to make sure.
y = 2(x) - 9
1 = 2(5) - 9
1 = 10 - 9
1 = 1, checks out.
So, the answer is y = 2x - 9.
4. This is the same concept as the previous question.
m = (y2 - y1) / (x2 - x1)
x1 = 14, y1 = 5, x2= 2, y2= 11
so we have m = (11 - 5) / (2 - 14) = (6) / (-12) = -1/2
Use the equation y = mx + b and the given points and calculated slope to solve for b.
5 = (-1/2)(14) + b
5 = -7 + b
b = 12
This gives us an equation y = (-1/2)x + 12
Check with the other point too, to make sure.
y = (-1/2)x + 12
11 = (-1/2)(2) + 12
11 = -1 + 12
11 = 11, checks ot.
So, the answer is y = (-1/2)x + 12.
5. This is also the same concept as #3.
m = (y2 - y1) / (x2 - x1)
= (-7 - 3) / (-2 - (-7))
= (-10) / (5)
= -2
Solve for b using y = mx + b with the given points and calculated slope.
3 = -2(-7) + b
3 = 14 + b
b = -11
This gives us an equation y = -2x - 11.
Check this with the other point too, just to make sure.
y = -2x - 11
-7 = -2(-2) - 11
-7 = 4 - 11
-7 = -7, checks out.
So, the answer is y = -2x -11.
6. This is also the same concept as #3.
m = (y2 - y1) / (x2 - x1)
= (8-7) / (6-3)
= 1/3
Use y = mx + b and the given points and calculated slope to solve for b.
7 = (1/3)(3) + b
7 = 1 + b
b = 6
So, this gives us an equation y = (1/3)x + 6.
Check with the other given point, just to make sure.
y = (1/3)x + 6
8 = (1/3)(6) + 6
8 = 2 + 6
8 = 8, checks out.
So, the answer is y = (1/3)x + 6.
Hope this helps!
