Solving Equations With Fractions Worksheets Pack - Twinkl - Free Printable

Let’s solve the problem step by step.

We are given two equations:

1) \( \frac{3}{x} + \frac{2}{y} = 5 \)

2) \( \frac{6}{x} - \frac{1}{y} = 4 \)

Our goal is to find the values of \( x \) and \( y \).

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Step 1: Let’s make substitution to simplify.

Let \( a = \frac{1}{x} \) and \( b = \frac{1}{y} \)

Then the equations become:

Equation (1): \( 3a + 2b = 5 \)

Equation (2): \( 6a - b = 4 \)

Now we have a system of linear equations in terms of \( a \) and \( b \). Much easier!

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Step 2: Solve the system.

We can use elimination or substitution. Let’s use elimination.

From Equation (2): \( 6a - b = 4 \)

Multiply this entire equation by 2 so that the coefficient of \( b \) becomes -2, which will cancel with +2b from Equation (1):

→ \( 2*(6a - b) = 2*4 \)

→ \( 12a - 2b = 8 \) → Call this Equation (3)

Now add Equation (1) and Equation (3):

Equation (1): \( 3a + 2b = 5 \)

Equation (3): \( 12a - 2b = 8 \)

Add them:

\( (3a + 12a) + (2b - 2b) = 5 + 8 \)

→ \( 15a = 13 \)

→ \( a = \frac{13}{15} \)

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Step 3: Plug back to find \( b \)

Use Equation (2): \( 6a - b = 4 \)

Plug in \( a = \frac{13}{15} \):

→ \( 6 * \frac{13}{15} - b = 4 \)

Calculate \( 6 * \frac{13}{15} = \frac{78}{15} = \frac{26}{5} \)

So:

→ \( \frac{26}{5} - b = 4 \)

Subtract \( \frac{26}{5} \) from both sides:

→ \( -b = 4 - \frac{26}{5} \)

Convert 4 to fifths: \( 4 = \frac{20}{5} \)

→ \( -b = \frac{20}{5} - \frac{26}{5} = -\frac{6}{5} \)

Multiply both sides by -1:

→ \( b = \frac{6}{5} \)

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Step 4: Recall what \( a \) and \( b \) represent.

We set:

\( a = \frac{1}{x} \), so \( x = \frac{1}{a} = \frac{1}{\frac{13}{15}} = \frac{15}{13} \)

\( b = \frac{1}{y} \), so \( y = \frac{1}{b} = \frac{1}{\frac{6}{5}} = \frac{5}{6} \)

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Step 5: Check our answer in original equations.

First equation: \( \frac{3}{x} + \frac{2}{y} = 3 * \frac{13}{15} + 2 * \frac{6}{5} \)

= \( \frac{39}{15} + \frac{12}{5} = \frac{13}{5} + \frac{12}{5} = \frac{25}{5} = 5 \) ✔

Second equation: \( \frac{6}{x} - \frac{1}{y} = 6 * \frac{13}{15} - \frac{6}{5} = \frac{78}{15} - \frac{6}{5} = \frac{26}{5} - \frac{6}{5} = \frac{20}{5} = 4 \) ✔

Perfect! Both check out.

Final Answer:
x = 15/13, y = 5/6