using the substitution p=1/x, q=1/y, solve simultaneous equations
1/x-5/y=y
2/x+1/y=s
please could show all working
so, make the suggested substitution:
p - 5q = 1/q
2p + q = s
I think there's a typo in the 1st equation. Don't expect to see a quadratic.
Madness
To solve the simultaneous equations using the substitution method, we need to rewrite the equations in terms of the new variables p and q.
Let's start by solving the first equation:
1/x - 5/y = y
Substituting p = 1/x and q = 1/y:
1/p - 5/q = q
Now let's solve the second equation:
2/x + 1/y = s
Substituting p = 1/x and q = 1/y:
2/p + 1/q = s
We now have the following system of equations:
1/p - 5/q = q ...........(1)
2/p + 1/q = s ...........(2)
To eliminate p, we can multiply equation (1) by 2 and equation (2) by 1:
2/p - 10/q = 2q ...........(3)
2/p + 1/q = s ...........(4)
Now we can subtract equation (4) from equation (3) to eliminate p:
-10/q - 1/q = 2q - s
Combining like terms:
-11/q = 2q - s
Multiplying both sides by q to eliminate the denominator:
-11 = 2q^2 - sq
Rearranging the terms:
2q^2 - sq - 11 = 0
This is a quadratic equation in terms of q. We can solve it by applying the quadratic formula:
q = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 2, b = -s, and c = -11. Substituting these values:
q = (-(-s) ± √((-s)^2 - 4(2)(-11))) / (2(2))
Simplifying:
q = (s ± √(s^2 + 88)) / 4
Now that we have the value of q, we can substitute it back into one of the original equations to solve for p.
Using equation (1):
1/p - 5/q = q
Substituting q = (s ± √(s^2 + 88)) / 4:
1/p - 5 / ((s ± √(s^2 + 88)) / 4) = (s ± √(s^2 + 88)) / 4
To simplify this equation, we can multiply both sides by p:
1 - 5p / ((s ± √(s^2 + 88)) / 4) = (s ± √(s^2 + 88)) / 4
Next, we can multiply both sides by ((s ± √(s^2 + 88)) / 4) to get rid of the denominators:
((s ± √(s^2 + 88)) / 4) - 5p = (s ± √(s^2 + 88))
Rearranging the terms:
-5p = (s ± √(s^2 + 88)) - ((s ± √(s^2 + 88)) / 4)
Combining like terms:
-5p = (3s ± 3√(s^2 + 88)) / 4
Multiplying both sides by -4/3 to solve for p:
p = -(3s ± 3√(s^2 + 88)) / 20
Therefore, the solutions for p and q in terms of s are:
p = -(3s ± 3√(s^2 + 88)) / 20
q = (s ± √(s^2 + 88)) / 4
To solve the simultaneous equations using the substitutions p=1/x and q=1/y, we will follow these steps:
Step 1: Express the given equations in terms of p and q.
We start with the given equations:
1/x - 5/y = y (Equation 1)
2/x + 1/y = s (Equation 2)
Substituting p=1/x and q=1/y into the equations, we get:
1/(1/p) - 5/(1/q) = 1/q (Equation 3)
2/(1/p) + 1/(1/q) = s (Equation 4)
Simplifying Equation 3, we have:
p - 5q = 1/q
Simplifying Equation 4, we have:
2p + q = s
Step 2: Solve for p in terms of q.
From Equation 4, we can isolate p:
2p = s - q
p = (s - q)/2 (Equation 5)
Step 3: Substitute the value of p into Equation 3.
Substituting Equation 5 (p = (s - q)/2) into Equation 3, we have:
(s - q)/2 - 5q = 1/q
Multiplying through by 2q to eliminate the denominator, we get:
(s - q)q - 10q^2 = 2
Expanding the equation, we have:
sq - q^2 - 10q^2 = 2
-sq - 11q^2 = 2
Rearranging the terms, we get:
11q^2 + sq + 2 = 0 (Equation 6)
Step 4: Solve the quadratic equation for q.
Now, we have a quadratic equation in terms of q. We can solve it using any suitable method, such as factoring or the quadratic formula. Let's solve it using the quadratic formula.
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, the quadratic equation is 11q^2 + sq + 2 = 0, where a = 11, b = s, and c = 2. Substituting these values into the quadratic formula, we get:
q = (-s ± √(s^2 - 4*11*2)) / (2*11)
Simplifying,
q = (-s ± √(s^2 - 88)) / 22 (Equation 7)
Step 5: Substitute the value of q into Equation 5 to solve for p.
Substituting the value of q from Equation 7 into Equation 5, we can solve for p:
p = (s - ((-s ± √(s^2 - 88)) / 22))/2
Simplifying,
p = (2s + s ± √(s^2 - 88)) / 44
p = (3s ± √(s^2 - 88)) / 44 (Equation 8)
Step 6: Express x and y in terms of p and q.
Recall the substitutions:
p = 1/x
q = 1/y
Therefore,
x = 1/p
y = 1/q
Substituting Equations 8 and 7 into these expressions, we can write x and y in terms of p and q:
x = 1/(3s ± √(s^2 - 88)) / 44
y = 1/((-s ± √(s^2 - 88)) / 22)
Simplifying,
x = 44/(3s ± √(s^2 - 88))
y = 22/((-s ± √(s^2 - 88)))