Leah has x half-dollars and y nickels, while James has x quarters and y dimes. Both have a total of 3$ . Compute the ordered pair(x,y).

We have \begin{align*}

50x+5y&=300,\\
25x+10y&=300.
\end{align*}Multiplying the second equation by $2$ and subtracting it from the first equation, we find $$-15y=-300\Rightarrow y=20.$$Substituting this into the first equation, $$50x+100=300\Rightarrow x=4.$$So the answer is $\boxed{(4,20)}$.

.50x+.05y=3.00

.25x+.1y=3.00

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Ah, Leah and James, the dynamic duo of pocket change. Let's crunch some numbers and reveal the glorious ordered pair (x, y).

Now, Leah has x half-dollars and y nickels. Since half-dollars are worth 50 cents and nickels are worth 5 cents, the total value of Leah's coins can be expressed as 50x + 5y cents.

On the other side of the coin (pun intended), James has x quarters and y dimes. Quarters are worth 25 cents each, and dimes are worth 10 cents each. So, the total value of James' coins is 25x + 10y cents.

Since both Leah and James have a total of $3, or 300 cents, we can set up the equation:

50x + 5y + 25x + 10y = 300

Simplifying this equation, we get:

75x + 15y = 300

Dividing both sides by 15:

5x + y = 20

Aha! We have a system of equations. Unfortunately, since we don't have any further information, we can't find a unique solution for x and y.

But worry not! We can still express the infinite possibilities through our clownish creativity. So, let's imagine one possibility: x = 4 and y = 0.

Therefore, the ordered pair (x, y) is (4, 0). Leah would have 4 half-dollars and no nickels, while James would have 4 quarters and no dimes.

Remember, this is just one possibility among many, so feel free to explore the comical combinations of x and y that satisfy the equation!

To find the ordered pair (x, y), we need to set up a system of equations based on the given information.

Let's start with Leah's information:
- Leah has x half-dollars: Each half-dollar is worth 50 cents or 0.50 dollars. So, Leah has a total value of 0.50x dollars.
- Leah also has y nickels: Each nickel is worth 5 cents or 0.05 dollars. So, Leah has a total value of 0.05y dollars.

Now, let's move to James's information:
- James has x quarters: Each quarter is worth 25 cents or 0.25 dollars. So, James has a total value of 0.25x dollars.
- James also has y dimes: Each dime is worth 10 cents or 0.10 dollars. So, James has a total value of 0.10y dollars.

Both Leah and James have a total of 3 dollars, so we can set up the following system of equations:

0.50x + 0.05y = 3 ... Equation 1
0.25x + 0.10y = 3 ... Equation 2

To solve this system of equations, we can use any method such as substitution or elimination.

Let's solve by elimination method:

Multiply Equation 1 by 20 to remove the decimals:
10x + y = 60 ... Equation 3

Now, subtract Equation 2 from Equation 3:
10x + y - (0.25x + 0.10y) = 60 - 3
9.75x + 0.9y = 57

Simplify:
975x + 9y = 5700 ... Equation 4

Now, we have a new equation (Equation 4) in terms of x and y.

To solve this equation, rearrange it to isolate one variable (y):
9y = 5700 - 975x
y = (5700 - 975x) / 9

Now, we can substitute this expression for y in Equation 1 and solve for x:

0.50x + 0.05[(5700 - 975x) / 9] = 3

Multiply through by 9 to remove the fraction:
4.5x + 0.05(5700 - 975x) = 27

Distribute:
4.5x + 285 - 48.75x = 27

Combine like terms:
-44.25x + 285 = 27

Subtract 285 from both sides:
-44.25x = -258

Divide both sides by -44.25:
x = 5.84 (approximately)

Now, substitute the value of x into Equation 3 to solve for y:

10(5.84) + y = 60
58.4 + y = 60
y = 60 - 58.4
y = 1.6 (approximately)

Therefore, the ordered pair (x, y) is approximately (5.84, 1.6).