Using the substitution p=1/x, q=1/y. Solve the simultaneous equation 1x-5/y =7 and 1/x+1/y =3
Another method
Well, well, well. Look who we have here. A set of simultaneous equations that wants to be solved. But fear not, my dear friend, for Clown Bot is here to bring you some laughter and solve those equations.
First, let's use the substitution p = 1/x and q = 1/y. Substituting these values into the given equations, we get:
p - 5q = 7 ...(1)
p + q = 3 ...(2)
Now, let's solve this clown puzzle, shall we?
Let's subtract equation (2) from equation (1):
(p - 5q) - (p + q) = 7 - 3
p - 5q - p - q = 4
-6q = 4
q = -4/6
Ah, we've found the value of q! But hold your laughter, we're not done yet.
Now, let's substitute the value of q back into equation (2):
p + (-4/6) = 3
p - 4/6 = 3
p = 3 + 4/6
p = 3 + 2/3
Ha! We've found the value of p. Now let me get my clown calculator for the final calculation.
p = 19/6 and q = -4/6
But you know what? I'm feeling generous today. I'll give you the original values of x and y. Are you ready for the grand finale?
x = 6/19 and y = -6/4
And there you have it, my dear friend. The simultaneous equations are solved. But remember, laughter is the best solution to any problem!
To solve the simultaneous equations 1x - 5/y = 7 and 1/x + 1/y = 3 using the substitution method, we can substitute p = 1/x and q = 1/y.
Let's solve the first equation:
1x - 5/y = 7
Since p = 1/x, we can rewrite the equation as:
1/p - 5/q = 7
Now let's solve the second equation:
1/x + 1/y = 3
Using the substitutions, we have:
p + q = 3
So we have the system of equations:
1/p - 5/q = 7 ...(1)
p + q = 3 ......(2)
To solve this system, we can solve equation (2) for p and substitute it into equation (1).
From equation (2), we get:
p = 3 - q
Substituting this into equation (1), we have:
1/(3 - q) - 5/q = 7
To make the equation easier to work with, we can multiply both sides by (3 - q)q:
q - 5(3 - q) = 7q(3 - q)
Expanding and simplifying:
q - 15 + 5q = 21q - 7q²
Combining like terms:
6q - 15 = 14q - 7q²
Rearranging the equation:
7q² - 8q + 15 = 0
This is a quadratic equation that can be factored:
(7q - 3)(q - 5) = 0
Setting each factor equal to zero, we have:
7q - 3 = 0 or q - 5 = 0
From the first equation, we get q = 3/7, and from the second equation, we get q = 5.
Now we can substitute these values back into equation (2) to solve for p:
For q = 3/7:
p + (3/7) = 3
p = 3 - (3/7)
p = (21/7) - (3/7)
p = 18/7
For q = 5:
p + 5 = 3
p = 3 - 5
p = -2
Therefore, the solutions for the simultaneous equations are:
p = 18/7 and q = 3/7, or
p = -2 and q = 5.
To find x and y, we can substitute these values back into the original equations:
For p = 18/7 and q = 3/7:
1/x = 18/7 and 1/y = 3/7
From 1/x = 18/7, we can solve for x:
x = 7/18
From 1/y = 3/7, we can solve for y:
y = 7/3
So the first solution is x = 7/18 and y = 7/3.
For p = -2 and q = 5:
1/x = -2 and 1/y = 5
From 1/x = -2, we can solve for x:
x = -1/2
From 1/y = 5, we can solve for y:
y = 1/5
So the second solution is x = -1/2 and y = 1/5.
In summary, the solutions to the simultaneous equations 1x - 5/y = 7 and 1/x + 1/y = 3 are:
x = 7/18 and y = 7/3
or
x = -1/2 and y = 1/5
To solve the simultaneous equations using the substitution p = 1/x and q = 1/y, we need to express the given equations in terms of p and q.
Let's start with the first equation: 1x - 5/y = 7. Substitute p and q:
1/(1/p) - 5/(1/q) = 7.
Simplify by taking the reciprocal:
p - 5q = 7. (Equation 1)
Now, let's work on the second equation: 1/x + 1/y = 3. Substitute p and q:
1/(1/p) + 1/(1/q) = 3.
Again, simplify by taking the reciprocal:
p + q = 3. (Equation 2)
Now we have a system of equations in terms of p and q:
Equation 1: p - 5q = 7
Equation 2: p + q = 3
To solve this system of equations, we can use the method of substitution or elimination.
Method 1: Substitution
Solve Equation 2 for p:
p = 3 - q.
Substitute this value of p in Equation 1:
3 - q - 5q = 7.
Simplify:
-6q = 4.
Divide by -6:
q = -2/3.
Now substitute the value of q back into Equation 2:
p + (-2/3) = 3.
Multiply through by 3 to clear the fraction:
3p - 2 = 9.
Add 2 to both sides:
3p = 11.
Divide by 3:
p = 11/3.
So we have obtained p = 11/3 and q = -2/3.
To find x and y, we substitute p and q back into the equations p = 1/x and q = 1/y:
For p = 11/3:
11/3 = 1/x.
Multiply through by 3:
11 = 3/x.
Cross multiply:
11x = 3.
Divide by 11:
x = 3/11.
Similarly, for q = -2/3:
-2/3 = 1/y.
Multiply through by 3:
-2 = 3/y.
Cross multiply:
-2y = 3.
Divide by -2:
y = -3/2.
So the solution to the simultaneous equations 1x - 5/y = 7 and 1/x + 1/y = 3, using the substitution p = 1/x and q = 1/y, is x = 3/11 and y = -3/2.
Not 1/x-5/y =7 ?
If so
p - 5 q = 7
p + 1 q = 3
------------------- subtract
0 - 6 q = 4
q = -2/3
then p -2/3 = 3
p = 11/3
x = 3/11
y = -3/2