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Isolating the variable in an equation is the process of getting the variable by itself on one side of the equation. When you encounter an equation like \( \frac{7}{8} c + w = 9 \), the main goal is to have \( c \) stand alone on one side. This makes it easy to identify what \( c \) equals and is crucial for solving equations.To start, you perform operations that will separate \( c \) from any other numbers or variables. In our example, we subtract \( w \) from both sides of the equation. This removes \( w \) from the left side, resulting in \( \frac{7}{8} c = 9 - w \). Isolating variables involves applying the same operation to both sides of the equation, ensuring the equation remains balanced.Once \( c \) is isolated, it's easier to complete the solution. This is a fundamental step in algebra and sets up the problem for any further algebraic manipulation needed.

Algebraic manipulation involves rearranging an equation to make it easier to work with. After isolating the term containing \( c \) in \( \frac{7}{8} c = 9 - w \), we need to "manipulate" it to solve for \( c \). This often includes operations such as addition, subtraction, multiplication, and division.In our case, we want to get rid of the fraction \( \frac{7}{8} \). This requires multiplying both sides of the equation by the reciprocal of \( \frac{7}{8} \), which is \( \frac{8}{7} \). This effectively "cancels out" the fraction on the left:

• Multiply the left side: \( \frac{8}{7} \times \frac{7}{8} c \equiv c \)
• Apply the same to the right side: \( \frac{8}{7} \times (9 - w) \)

Through these manipulations, we've transformed the equation into \( c = \frac{8}{7} (9 - w) \). Understanding how to manipulate equations through these operations is a key skill in algebra.

Fraction elimination is the process of removing fractions from an equation to simplify it. Fractions can complicate equations, making them harder to solve quickly. In solving \( \frac{7}{8} c = 9 - w \), eliminating the fraction is critical to finding \( c \).The goal is to clear the fraction, which we do by multiplying by the reciprocal. The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). This means for \( \frac{7}{8} \), the reciprocal is \( \frac{8}{7} \). By multiplying both parts of the equation by \( \frac{8}{7} \), the fraction on the left becomes 1, simplifying to:

• \( \frac{8}{7} \times \frac{7}{8} c = c \)
• On the right: \( \frac{8}{7} \times (9 - w) = \frac{8}{7}(9 - w) \)

This gives us the final expression \( c = \frac{8}{7}(9 - w) \). Conquering fraction elimination is very helpful as it often simplifies solving and interpreting equations.