find the value of y so that the line passing through the two points has the given slope: 1. (6,y), (3/2, 9/5), m=2/3
2. (12,y), (-6,9/7), m=-1/6
Let's take the first one. We know that any line can be expressed as
y = mx + c
First we plug in the m given:
y = (2/3)x + c.
Now, we know that the point (3/2, 9/5) is on this line, so we can substitute these values into our equation to find c:
9/5 = (2/3)(3/2) + c
9/5 = 1 + c
c = 4/5
So our line is
y = (2/3)x + 4/5
Knowing that, you should be able to find the y value of the point at x = 6, to find the point (6, y).
And the second one is just like it.
To find the value of y for the first question, we can use the formula for slope:
m = (y₂ - y₁) / (x₂ - x₁)
Using the points (6, y) and (3/2, 9/5), and the given slope m = 2/3, we can plug in the values into the slope formula:
2/3 = (9/5 - y) / (3/2 - 6)
Next, we can simplify the equation:
2/3 = (9/5 - y) / (-9/2)
To get rid of the fractions, we can cross-multiply:
2(-9/2) = 3(9/5 - y)
-9 = 27/5 - 3y
Now, we can solve for y:
-9 - 27/5 = -3y
(-45/5) - (27/5) = -3y
-72/5 = -3y
Divide both sides by -3 to isolate the y variable:
(-72/5) / (-3) = y
24/5 = y
Therefore, the value of y is 24/5 for the line passing through the points (6, y) and (3/2, 9/5) with a slope of 2/3.
For the second question, we can follow a similar process.
Using the points (12, y) and (-6, 9/7), and the given slope m = -1/6, we can plug in the values into the slope formula:
-1/6 = (9/7 - y) / (-6 - 12)
Simplifying the equation:
-1/6 = (9/7 - y) / (-18)
Cross-multiplying:
-1(-18) = 6(9/7 - y)
18 = 54/7 - 6y
Now, we can solve for y:
18 - 54/7 = -6y
(126/7 - 54/7) = -6y
(72/7) = -6y
Divide both sides by -6 to isolate the y variable:
(72/7) / (-6) = y
-12/7 = y
Therefore, the value of y is -12/7 for the line passing through the points (12, y) and (-6, 9/7) with a slope of -1/6.
To find the value of y for each line, we can use the slope-intercept form of a linear equation, which is y = mx + b. In this form, m represents the slope of the line, and b represents the y-intercept.
Let's solve the first question:
1. (6, y), (3/2, 9/5), m = 2/3
To find the value of y, we can substitute the values of one of the points, along with the given slope, into the slope-intercept form equation and solve for y.
Using the point (6, y):
y = mx + b
y = (2/3)(6) + b
y = 4 + b
Now, we can use the other point (3/2, 9/5) to find the y-intercept, b.
9/5 = (2/3)(3/2) + b
9/5 = 1 + b
b = 9/5 - 1
b = 9/5 - 5/5
b = 4/5
Now that we've found the value of b, we can substitute it back into the equation:
y = mx + b
y = (2/3)(6) + 4/5
y = 12/3 + 4/5
y = 4 + 4/5
y = 24/5 + 4/5
y = 28/5
y = 5.6
Therefore, the value of y for the line passing through the points (6, y) and (3/2, 9/5) with a given slope of 2/3 is y = 5.6.
Now, let's solve the second question:
2. (12, y), (-6, 9/7), m = -1/6
Following the same steps as before, let's use the point (12, y):
y = mx + b
y = (-1/6)(12) + b
y = -2 + b
Using the other point (-6, 9/7) to find the y-intercept, b:
9/7 = (-1/6)(-6) + b
9/7 = 1 + b
b = 9/7 - 7/7
b = 2/7
Substituting the value of b back into the equation:
y = mx + b
y = (-1/6)(12) + 2/7
y = -2 + 2/7
y = -14/7 + 2/7
y = -12/7
y = -1.714
Therefore, the value of y for the line passing through the points (12, y) and (-6, 9/7) with a given slope of -1/6 is y = -1.714.