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Solve the system of equations:
[tex]\[ \frac{x}{4} + \frac{y}{5} = 22 \][/tex]
[tex]\[ \frac{x}{5} + \frac{y}{4} - 1 = 22 \][/tex]

Sagot :

GinnyAnswer
Certainly! Let's solve the system of equations step by step:

The given system of equations is:

1) [tex]\(\frac{x}{4} + \frac{y}{5} = 22\)[/tex]

2) [tex]\(\frac{x}{5} + \frac{y}{4} - 1 = 22\)[/tex]

First, we simplify the second equation:

[tex]\(\frac{x}{5} + \frac{y}{4} - 1 = 22\)[/tex]

Adding 1 to both sides, we get:

[tex]\(\frac{x}{5} + \frac{y}{4} = 23\)[/tex]

Now, we have the following system of equations:

1) [tex]\(\frac{x}{4} + \frac{y}{5} = 22\)[/tex]

2) [tex]\(\frac{x}{5} + \frac{y}{4} = 23\)[/tex]

Let's rewrite these equations in a form that eliminates the fractions by multiplying through by the common denominators.

For the first equation:

Multiply everything by 20 (the least common multiple of 4 and 5):

[tex]\(20 \cdot \left(\frac{x}{4}\right) + 20 \cdot \left(\frac{y}{5}\right) = 20 \cdot 22\)[/tex]

This simplifies to:

[tex]\(5x + 4y = 440\)[/tex] ...(3)

For the second equation:

Multiply everything by 20 (the least common multiple of 5 and 4):

[tex]\(20 \cdot \left(\frac{x}{5}\right) + 20 \cdot \left(\frac{y}{4}\right) = 20 \cdot 23\)[/tex]

This simplifies to:

[tex]\(4x + 5y = 460\)[/tex] ...(4)

So now we have a new system of linear equations:

3) [tex]\(5x + 4y = 440\)[/tex]

4) [tex]\(4x + 5y = 460\)[/tex]

Next, we solve this system using the elimination method.

Multiply equation (3) by 4:

[tex]\(20x + 16y = 1760\)[/tex] ...(5)

Multiply equation (4) by 5:

[tex]\(20x + 25y = 2300\)[/tex] ...(6)

Now, subtract equation (5) from equation (6):

[tex]\((20x + 25y) - (20x + 16y) = 2300 - 1760\)[/tex]

[tex]\(20x + 25y - 20x - 16y = 540\)[/tex]

[tex]\(9y = 540\)[/tex]

Divide by 9:

[tex]\(y = 60\)[/tex]

Now substitute [tex]\(y = 60\)[/tex] back into equation (3) to find [tex]\(x\)[/tex]:

[tex]\(5x + 4(60) = 440\)[/tex]

[tex]\(5x + 240 = 440\)[/tex]

Subtract 240 from both sides:

[tex]\(5x = 200\)[/tex]

Divide by 5:

[tex]\(x = 40\)[/tex]

Therefore, the solution to the system of equations is:

[tex]\(x = 40\)[/tex] and [tex]\(y = 60\)[/tex].
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