To solve the system of equations using the elimination method, the goal is to eliminate one of the variables by adding or subtracting the equations. Here's how you can approach it, even though the equations contain fractions:

The system of equations is:

\frac{x}{11} - \frac{y}{8} = -2 \tag{1} \frac{x}{2} - \frac{y}{4} = 3 \tag{2}

Step-by-Step Process

1. Eliminate Fractions by Multiplying:

   To eliminate the fractions, multiply each equation by the least common denominator (LCD) of the denominators in each equation.

   • For equation (1), the denominators are 11 and 8. The LCD of 11 and 8 is 88. Multiply the entire equation by 88: $88 \times \left(\frac{x}{11} - \frac{y}{8}\right) = 88 \times (-2)$ Simplifying: 8x - 11y = -176 \tag{3}

   • For equation (2), the denominators are 2 and 4. The LCD of 2 and 4 is 4. Multiply the entire equation by 4: $4 \times \left(\frac{x}{2} - \frac{y}{4}\right) = 4 \times 3$ Simplifying: 2x - y = 12 \tag{4}

2. Align Equations for Elimination:

   The two equations are now: 8x - 11y = -176 \tag{3} 2x - y = 12 \tag{4}

3. Eliminate one of the variables:

   Let's eliminate $y$. To do this, multiply equation (4) by 11 to match the coefficient of $y$ in equation (3): $11(2x - y) = 11(12)$ This gives: 22x - 11y = 132 \tag{5}

4. Subtract the equations:

   Now subtract equation (3) from equation (5) to eliminate $y$: $(22x - 11y) - (8x - 11y) = 132 - (-176)$ Simplifying: $14x = 308$ Solve for $x$: $x = \frac{308}{14} = 22$

5. Substitute $x = 22$ into one of the original equations:

   Now, substitute $x = 22$ into equation (4): $2(22) - y = 12$ Simplifying: $44 - y = 12$ Solve for $y$: $y = 44 - 12 = 32$

Final Answer:

The solution is: $x = 22, \quad y = 32$ Thus, the point of intersection is $(22, 32)$.

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Would you like further details or clarification on any step?

Related Questions:

1. How do you solve systems of equations using substitution?
2. What are the advantages of the elimination method over substitution?
3. How do you determine if a system of equations has no solution or infinitely many solutions?
4. Can you solve a system of nonlinear equations by elimination?
5. How can you graphically verify the solution of a system of equations?

Tip:

When dealing with fractions, always try to eliminate them by multiplying through by the least common denominator. This simplifies the arithmetic and makes elimination or substitution easier!