The variables x and y are related by the equation y=ax/(x-b),where a and b are constants .When y is plotted against y/x ,a straight line is obtained .The line has gradient 3 and y-intercept 2.
(i)Find the values of a and b
(ii)Find the values of x which x=y
(i) To find the values of a and b, we can use the equation of the line: y = mx + c, where m is the gradient and c is the y-intercept.
Given that the gradient is 3 and the y-intercept is 2, we have:
y = 3x + 2
Now, let's compare this equation with the given equation: y = ax/(x - b)
We can see that a = 3 and b = -2.
Therefore, the values of a and b are 3 and -2, respectively.
(ii) To find the values of x when x = y, we can substitute x = y into the equation y = 3x + 2:
y = 3(x) + 2
y = 3x + 2
Now, substitute x = y:
y = 3(y) + 2
y = 3y + 2
Rearrange the equation:
2 = 2y
y = 1
Now, substitute y = 1 back into the original equation:
1 = 3x + 2
Rearrange the equation to solve for x:
3x = -1
x = -1/3
Therefore, the values of x when x = y are x = -1/3 and y = 1.
To find the values of a and b, we can use the given information about the gradient and y-intercept of the straight line.
(i) Gradient-Intercept Form:
The equation of a straight line in gradient-intercept form is given by y = mx + c, where m represents the gradient, and c represents the y-intercept.
In this case, we know that the line has a gradient of 3 and a y-intercept of 2. So, we can equate these values:
m = 3
c = 2
Comparing this with the given equation y = ax/(x - b), we can see that m = a and c = ab.
Therefore,
a = 3
ab = 2
To solve for b, we can divide ab = 2 by a = 3:
b = 2/3
So, the values of a and b are a = 3 and b = 2/3.
(ii) Finding the values of x when x = y:
We will substitute x = y into the given equation y = ax/(x - b) and solve for x.
Replacing x with y, we get:
y = a * y / (y - b)
Multiplying both sides by (y - b), we get:
y(y - b) = a * y
Expanding the equation:
y^2 - by = ay
Rearranging the terms:
y^2 - ay - by = 0
Now, we can substitute the value of a = 3 into the equation:
y^2 - 3y - (2/3)y = 0
Multiplying through by 3 to eliminate fractions:
3y^2 - 9y - 2y = 0
Combining like terms:
3y^2 - 11y = 0
Factoring out y:
y(3y - 11) = 0
So, either y = 0 or 3y - 11 = 0.
If y = 0, then x = 0 as well because x = y. So, (x,y) = (0,0) is a solution.
If 3y - 11 = 0, then y = 11/3. And since x = y, we have x = 11/3 as well. So, another solution is (x,y) = (11/3, 11/3).
Therefore, the values of x which satisfy x = y are x = 0 and x = 11/3.
To find the values of a and b in the equation y = ax/(x - b), we can use the given information that when y is plotted against y/x, a straight line with a gradient of 3 and a y-intercept of 2 is obtained.
(i) Find the values of a and b:
Let's start by rearranging the equation y = ax/(x - b) to isolate a and b separately.
Multiply both sides of the equation by (x - b) to eliminate the denominator:
y(x - b) = ax
Expand the left side:
yx - yb = ax
Rearrange the equation to bring all terms with x on one side:
ax - yx = yb
Factor out x:
x(a - y) = yb
Divide both sides of the equation by (a - y):
x = (yb) / (a - y)
Now, we have the equation x = (yb) / (a - y), which relates x and y.
The question states that when y is plotted against y/x, a straight line with a gradient of 3 and a y-intercept of 2 is obtained. This means that the equation y/x = mx + c holds, where m is the gradient and c is the y-intercept.
In this case, we have:
y/x = 3x + 2
Multiplying both sides of the equation by x to eliminate the denominator:
y = 3x^2 + 2x
Comparing this with the equation x = (yb) / (a - y), we can see that the coefficients of x must be the same. Therefore, we have:
a - y = 2
Substituting this value back into the equation x = (yb) / (a - y):
x = (yb) / 2
Now, we can substitute x and y from the given equation:
(3x^2 + 2x)b / 2 = x
Multiply both sides of the equation by 2/(3x^2 + 2x) to eliminate the denominator:
b = 2x / (3x^2 + 2x)
Since we are looking for the values of a and b, we need to solve the simultaneous equations:
a - y = 2
b = 2x / (3x^2 + 2x)
(ii) Find the values of x when x = y:
To find the values of x when x = y, we substitute x for y in the equation y = ax/(x - b):
x = ax / (x - b)
Multiply both sides of the equation by (x - b) to eliminate the denominator:
x(x - b) = ax
Expand the left side:
x^2 - bx = ax
Rearrange the equation to bring all terms with x on one side:
x^2 - bx - ax = 0
Factor out x:
x(x - b - a) = 0
Since x cannot be equal to 0, we can solve (x - b - a) = 0:
x - b - a = 0
Solve for x:
x = b + a
Therefore, the values of x when x = y are x = b + a.
y/x = a/(x-b)
the given line has equation (slope-intercept)
y = 2 + 3(y/x)
xy = 2x + 3y
y = 2x/(x-3)
so, a=2 b=3
if x=y, y = 2+3 = 5
check: 5 = 2*5/(5-3)