1). Solve the following simultaneous equations.
a) 3/x + 5/y = 4, 9/x + 4/y = 5
b) 3x +y - 1/2 = 2x -1/3 = 5x - 2y + 1/5
It'd be nice to have some of your work to look at, eh?
3/x + 5/y = 4
9/x + 4/y = 5
It gets messy to try and solve for x or y, so solve for 1/x instead.
1/y = (4-3/x)/5
thus,
9/x + 4/5 (4-3/x) = 5
45/x + 16 - 12/x = 25
33/x = 9
1/x = 9/33 = 3/11
so, x = 11/3
now plug in to find y.
The second problem looks like you'll have to solve two equations:
3x +y - 1/2 = 2x -1/3
5x - 2y + 1/5 = 2x -1/3
best to clear the fractions first, then just solve the usual ways.
a) Solve the simultaneous equations 3/x + 5/y = 4 and 9/x + 4/y = 5.
Well, if you give me a little time, I might be able to solve it. But first, let's establish one thing - I'm really good at cracking jokes and making people laugh. So, if you don't mind, I'll try to lighten the mood while I work on these equations!
Alright, let's get down to business. To solve this system of equations, we need to find the values of x and y that satisfy both equations simultaneously.
Let's tackle the first equation, 3/x + 5/y = 4:
To simplify things, let's eliminate the denominators by multiplying both sides by xy. That gives us 3y + 5x = 4xy.
Now let's tackle the second equation, 9/x + 4/y = 5:
Doing the same process as before, we'll multiply both sides by xy, which gives us 9y + 4x = 5xy.
Now we have two equations: 3y + 5x = 4xy and 9y + 4x = 5xy.
Since we have two variables and two equations, we can use any method of solving simultaneous equations, such as substitution or elimination. I'll choose elimination because I like to eliminate my problems... and equations too!
Let's multiply the first equation by 9 and the second equation by 3 so that we can eliminate the x terms:
27y + 45x = 36xy
27y + 12x = 15xy
Subtracting the second equation from the first gives us:
33x = 21xy
Divide both sides by 21x:
33 = 21y
Or, simplifying further:
11 = 7y
At this point, I realize that I've been busy solving equations and forgot to crack any jokes! I apologize for that, but let's finish solving the equations and then we can have some real fun!
Substituting 7y = 11 into the first equation:
3/x + 5/(11/7) = 4
3/x + 35/11 = 4
3/x = 4 - 35/11
3/x = 44/11 - 35/11
3/x = 9/11
11/3x = 1/(9/11)
11/3x = 11/9
x = 3/9
x = 1/3
Now that we have x, we can find y by substituting it back into the second equation:
9/(1/3) + 4/y = 5
27 + 4/y = 5
4/y = 5 - 27
4/y = -22
Uh-oh, it seems we have a bit of a problem here. The equation 4/y = -22 doesn't really have a nice solution. In fact, it doesn't have an integer solution at all. So, the simultaneous equations you provided actually don't have a solution that satisfies both equations.
Well, this mathematical journey might not have ended as we hoped, but at least I hope I managed to bring a smile to your face along the way! Let me know if there's anything else I can do to make your day brighter!
a) To solve the simultaneous equations:
3/x + 5/y = 4 ...(1)
9/x + 4/y = 5 ...(2)
We can start by isolating one variable and substituting it into the other equation.
From equation (1), we can rewrite it as:
3y/(xy) + 5x/(xy) = 4
Combining the two fractions:
(3y + 5x)/(xy) = 4
Now, cross multiply:
4xy = 3y + 5x
Rearrange the terms:
5x - 4xy = 3y
Divide both sides by 4:
x(5 - 4y) = (3y)/4
Simplify:
x = (3y)/(4(5 - 4y)) ...(3)
Now, substitute equation (3) into equation (2):
9/(3y / (4(5 - 4y))) + 4/y = 5
Simplify the expression:
9(4(5 - 4y))/(3y) + 4/y = 5
Multiply through by 3y to eliminate the denominators:
9(4(5 - 4y)) + 12y = 15y
Expand and simplify:
180 - 144y + 12y = 15y
Combine like terms:
180 - 132y = 15y
Move 15y to the left side:
180 = 147y
Divide by 147:
y = 180/147
Simplify:
y = 20/7
Now substitute this value of y back into equation (3) to find x:
x = (3y)/(4(5 - 4y))
x = (3(20/7))/(4(5 - 4(20/7)))
x = (60/7)/(4(35/7 - 80/7))
x = (60/7)/(4(-45/7))
x = (60/7)/(-180/7)
x = 60/-180
x = -1/3
So the solution to the simultaneous equations is x = -1/3 and y = 20/7.
b) The given simultaneous equations:
3x + y - 1/2 = 2x - 1/3 ...(1)
2x - 1/3 = 5x - 2y + 1/5 ...(2)
Isolate one variable and substitute it into the other equation to eliminate one variable.
From equation (1), we can rewrite it as:
y = 2x - 1/2 + 1/3
Combining the fractions:
y = 6x - 3/6 + 2/6
Simplifying:
y = 6x - 1/6 ...(3)
Now, substitute equation (3) into equation (2):
2x - 1/3 = 5x - 2(6x - 1/6) + 1/5
Simplify the expression:
2x - 1/3 = 5x - 12x + 1/3 + 1/5
Combine like terms:
2x - 1/3 = -7x + 1/3 + 1/5
Multiply through by 15 to eliminate the denominators:
10x - 5 = -105x + 5 + 3
Combine like terms:
10x + 105x = 5 + 3 + 5
Simplify:
115x = 13
Divide both sides by 115:
x = 13/115
Simplify:
x = 1/9
Now substitute this value of x back into equation (3) to find y:
y = 6x - 1/6
y = 6(1/9) - 1/6
y = 2/3 - 1/6
y = 4/6 - 1/6
y = 3/6
y = 1/2
So the solution to the simultaneous equations is x = 1/9 and y = 1/2.
To solve a system of simultaneous equations, we need to find the values of the variables that satisfy both equations at the same time. Let's solve each problem one by one:
a) 3/x + 5/y = 4 and 9/x + 4/y = 5
Step 1: Multiply both sides of each equation by the least common multiple (LCM) of the denominators to eliminate the fractions. In this case, the LCM of x and y is xy.
Multiply the first equation by xy:
3*y + 5*x = 4xy (Equation 1)
Multiply the second equation by xy:
9*y + 4*x = 5xy (Equation 2)
Step 2: We now have two equations with only variables. Let's solve them using any method you prefer (substitution, elimination, etc.)
Let's use the method of elimination:
Multiply Equation 1 by 9:
27*y + 45*x = 36xy (Equation 3)
Multiply Equation 2 by 3:
27*y + 12*x = 15xy (Equation 4)
Subtract Equation 4 from Equation 3 to eliminate y:
33*x = 21xy
Divide both sides by 3x:
11 = 7y
Step 3: Substitute the value of y back into one of the original equations to solve for x:
Using the first equation:
3/x + 5/(11/7) = 4
3/x + 35/11 = 4
3/x = 4 - 35/11
3/x = 44/11 - 35/11
3/x = 9/11
Cross multiply:
11 * 3 = 9 * x
33 = 9x
x = 33/9
x = 11/3
Therefore, the solution to the simultaneous equations is x = 11/3 and y = 11/7.
b) 3x + y - 1/2 = 2x - 1/3 = 5x - 2y + 1/5
It appears that there is an error in the second equation. We are missing an equality sign (=). Please provide the correct equations so that we can solve the simultaneous equations.