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When solving systems of equations, fractions can make the process more complicated. Thus, eliminating fractions makes equations easier to manage. To do this, you can multiply every term by the least common denominator (LCD). This step transforms fractions into whole numbers.

For instance, in the equation \( \frac{x-2}{8} + \frac{y+1}{2} = -6 \), notice the LCD is 8. By multiplying each term by 8, the equation becomes fraction-free: \ \( 8 \times \frac{x-2}{8} + 8 \times \frac{y+1}{2} = -6 \times 8 \ x-2 + 4(y+1) = -48 \ x + 4y + 2 = -48 \ x + 4y = -50 \ \).

Similarly, apply the same technique to the second equation. The objective is to rewrite equations in simpler formats to make onward steps, like substitution or elimination, more manageable.

Variable substitution involves replacing one variable with an equivalent expression involving the other variable. This method is effective after simplifying the equations.

First, solve one equation for one variable. Let's take \( x + 4y = -50 \) and rearrange it for \( x \): \ \( x = -50 - 4y \ \).

Next, substitute this expression into the other equation. Substituting \( x = -50 - 4y \) into \( 2x - y = 53 \) gives: \ \( 2(-50-4y) - y = 53 \ -100 - 8y - y = 53 \ -9y = 153 \ y = -17 \ \).

Substitution simplifies the remaining efforts to solve the system by reducing the number of variables.

Verification ensures that your solutions satisfy the original equations. This step is crucial because errors can occur during arithmetic manipulations.

To verify, substitute the found values back into the original equations. Our solutions are \( x = 18 \) and \( y = -17 \).

First Equation: \( \frac{18-2}{8} + \frac{-17+1}{2} = -6 \ \frac{16}{8} + \frac{-16}{2} = -6 \ 2 - 8 = -6 \ -6 = -6 \ \), which is true.

Second Equation: \( \frac{18-2}{2} - \frac{-17+1}{4} = 12 \ \frac{16}{2} - \frac{-16}{4} = 12 \ 8 + 4 = 12 \ 12 = 12 \ \), which also holds true.

Both verifications confirm that our solutions are correct.

Finding common denominators is essential for combining fractions effectively. The first step involves identifying the Least Common Denominator (LCD), the smallest number both denominators can divide.

For instance, in \( \frac{x-2}{8} + \frac{y+1}{2} \ \), 8 is the LCD between 8 and 2. Rewriting \( \frac{y+1}{2} \ \) as \( \frac{4(y+1)}{8} \ \) makes combining fractions straightforward: \ \( \frac{x-2 + 4(y+1)}{8} = -6 \ \).

The same method simplifies equations by dealing with one denominator.

In the second equation, \( \frac{x-2}{2} - \frac{y+1}{4} \ \), the LCD is 4. By rewriting and using the common denominator, we get: \ \( \frac{2(x-2)}{4} - \frac{y+1}{4} \ \), further simplifying helps to combine and solve easily. This makes managing equations systematically easier.