91Ó°ÊÓ

The substitution method is a common strategy for solving equations or verifying solutions to equations. This technique involves replacing a variable with a given value to see if it satisfies the equation.
In equations like the one in our example, you often have an unknown variable that you aim to solve for. The purpose of substitution is to test if a particular value for this unknown, when substituted back into the equation, makes the equation true.
In the given exercise, the question is whether \( x = \frac{1}{9} \) is a solution to the equation \( x - \frac{4}{7} = -\frac{31}{63} \).

• We substitute \( x \) with \( \frac{1}{9} \)
• The equation now becomes \( \frac{1}{9} - \frac{4}{7} = -\frac{31}{63} \)

By doing this, we can quickly verify if the results match both sides of the equation, thereby determining if \( \frac{1}{9} \) is indeed a solution.

When dealing with fractions, it's important to have a consistent way to perform operations like addition or subtraction. A common denominator simplifies this process by converting fractions into equivalent forms where their denominators are the same.
Finding the common denominator involves:

• Identifying the least common multiple (LCM) of the denominators involved.
• Converting each fraction to an equivalent fraction with this common denominator.

For the fractions \( \frac{1}{9} \) and \( \frac{4}{7} \), the LCM of 9 and 7 is 63. We convert:
\( \frac{1}{9} = \frac{7}{63} \) and \( \frac{4}{7} = \frac{36}{63} \)
This makes it easier to subtract the fractions as they are now like fractions with the same denominator.

Subtracting fractions is straightforward once they're expressing with a common denominator. This allows for easy comparison and arithmetical operations.
After gaining a common denominator, fraction subtraction involves:

• Subtracting the numerators.
• Keeping the common denominator unchanged.

In the exercise, after converting the fractions to \( \frac{7}{63} \) and \( \frac{36}{63} \), we subtract:
\( \frac{7}{63} - \frac{36}{63} = -\frac{29}{63} \)
Finally, this result is compared to the given right-hand side of the equation \( -\frac{31}{63} \). Since the results differ, \( \frac{1}{9} \) is not a solution to the equation.