A line with gradient of -3,passes through points 3.k and k.8 find the value of k and hence express the equation of the line in the form of ax+by=c ,where a,b and c are constants
Standard way to write points:
(3,k) and (k,8)
we know (8-k)/(k-3) = -3
8 - k = -3k + 9
2k = 1
k = 1/2
equation using m = -3, point is (3, 1/2)
y - 1/2 = -3(x - 3)
times 2
2y - 1 = -6(x-3)
2y - 1 = -6x + 18
6x + 2y = 19
Well, let's put on our clown shoes and solve this math problem with a little humor!
To find the value of k, we can use the formula for the gradient (m) of a line: m = (change in y) / (change in x).
In this case, the gradient is given as -3. So we have the equation:
-3 = (8 - k) / (3 - k)
But solving this equation isn't going to be as easy as squirting water out of a flower, so we'll have to cross multiply and distribute that clown makeup:
-3(3 - k) = (8 - k)
Alright, let's simplify this circus act:
-9 + 3k = 8 - k
It's time to group the k's and the numbers together:
4k = 17
Finally, we'll divide both sides by 4 to find out what k is:
k = 17/4
So, the value of k is 17/4. Now, let's move on to the second part of the question.
To express the equation of the line in the form of ax + by = c, we need to substitute the value of k back into the equation.
The coordinates are given as (3, k) and (k, 8). So, we can use the point-slope formula:
(y - y1) = m(x - x1)
Using the given gradient, the equation becomes:
(y - k) = -3(x - 3)
Expanding this equation, we get:
y - k = -3x + 9
Now, let's rearrange it in the form of ax + by = c:
3x + y = 9 + k
Since k = 17/4, we can substitute it in:
3x + y = 9 + (17/4)
Finally, we can simplify this equation by getting a common denominator:
3x + y = 36/4 + 17/4
Using our clown math skills, we sum up the fractions:
3x + y = 53/4
And there you have it, the equation of the line in the form of ax + by = c is 3x + y = 53/4. Now, let's bring in some circus music and celebrate solving the problem in a funny way! 🎪🎉🤡
To find the value of k, we can use the gradient formula. The formula for the gradient or slope of a line is given by:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
In this case, the line passes through the points (3, k) and (k, 8). So we can substitute these values into the formula:
-3 = (8 - k) / (k - 3)
To solve this equation, we can cross-multiply:
-3(k - 3) = 8 - k
-3k + 9 = 8 - k
Next, we can combine like terms:
-2k + 9 = 8
Subtract 9 from both sides:
-2k = 8 - 9
-2k = -1
Divide both sides by -2:
k = -1 / -2
k = 1/2
So the value of k is 1/2.
Next, to express the equation of the line in the form of ax + by = c, we need to find the values of a, b, and c.
We already know that the gradient or slope of the line is -3. The formula for the gradient-intercept form of a line is given by:
y = mx + b
where m is the gradient and b is the y-intercept.
Substituting the values, we have:
y = -3x + b
To find the value of b, we can substitute the coordinates (3, k) or (k, 8):
k = -3 * 3 + b
k = -9 + b
or
8 = -3 * k + b
8 = -3k + b
Now, we can substitute the value of k we found earlier (k = 1/2) into either of these equations to find the value of b.
Using the equation k = -9 + b:
1/2 = -9 + b
Adding 9 to both sides:
b = 9 + 1/2
b = 9 1/2
So, the value of b is 9 1/2.
Now, the equation of the line in the form of ax + by = c becomes:
y = -3x + 9 1/2
To express this equation with constants a, b, and c, we multiply every term by 2 to eliminate the fraction:
2y = -6x + 19
So, the equation of the line in the form of ax + by = c is:
6x + 2y = 19
To find the value of k and express the equation of the line in the form of ax + by = c, we can use the gradient-intercept form of a line, which is given by y = mx + c, where m is the gradient and c is the y-intercept.
Given that the gradient (m) of the line is -3, we can write the equation of the line in slope-intercept form as:
y = -3x + c
Now, we need to find the value of k to determine the y-intercept (c). We are given that the line passes through the points (3, k) and (k, 8).
Substituting x = 3 and y = k into the equation of the line, we get:
k = -3(3) + c
k = -9 + c
Similarly, substituting x = k and y = 8, we get:
8 = -3k + c
Now we have a system of two equations:
k = -9 + c
8 = -3k + c
We can solve this system of equations to find the value of k and c.
Subtract the first equation from the second equation to eliminate the c term:
8 - k = -3k + c - (-9 + c)
8 - k = -3k + 9
k - 3k = -1
-2k = -1
k = -1 / -2
k = 1 / 2
Now that we have the value of k, we can substitute it back into one of the original equations to find the value of c.
Using k = 1/2 in the equation k = -9 + c:
1/2 = -9 + c
c = 1/2 + 9
c = 9 1/2
So, the value of k is 1/2 and the value of c is 9 1/2.
Finally, we can express the equation of the line in the form of ax + by = c by substituting the values of k and c into y = -3x + c:
y = -3x + 9 1/2
To write the equation in the form ax + by = c, we need to eliminate fractions. Multiply the equation by 2 to get rid of the fraction:
2y = -6x + 19
Now, rearrange the equation to isolate the terms with x and y:
6x + 2y = 19
Therefore, the equation of the line in the form ax + by = c is 6x + 2y = 19.