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An algebraic equation is a mathematical statement that shows the equality of two expressions with a variable. In the given problem, we are working with the equation: \( \frac{2}{5} w - 6 = \frac{7}{5} w \). This is a linear algebraic equation since the variable \( w \) appears to the first power. Main parts of an algebraic equation include:

• Variable: A symbol that can represent different values (e.g., \( w \))
• Coefficient: A numerical or constant factor that multiplies the variable (e.g., \( \frac{2}{5} \))
• Constant: A fixed value that doesn’t change (e.g., -6)

The goal is to find the value of the variable that makes the equation true. In this problem, we will find the value of \( w \).

To solve the equation, we isolate the variable on one side. This means we want to get \( w \) by itself on one side of the equation. Here's how to isolate \( w \) in our problem: 1. Start with the original equation: \[ \frac{2}{5} w - 6 = \frac{7}{5} w \] 2. Subtract \( \frac{2}{5} w \) from both sides: \[ \frac{2}{5} w - 6 - \frac{2}{5} w = \frac{7}{5} w - \frac{2}{5} w \] 3. Simplify the terms: \[ -6 = \frac{5}{5} w \] 4. Since \( \frac{5}{5} \) is 1, this simplifies to: \[ -6 = w \] Now, we have isolated \( w \) and found the solution: \( w = -6 \). Isolating the variable involves moving all terms involving the variable to one side and constants to the other. Use operations that maintain the equality, such as addition, subtraction, multiplication, and division.

After solving the equation, it’s vital to verify that the solution is correct. Verification involves substituting the solution back into the original equation to check if both sides are equal. To verify that \( w = -6 \) is correct:1. Substitute \( w = -6 \) into the original equation: \[ \frac{2}{5}(-6) - 6 = \frac{7}{5}(-6) \] 2. Calculate the left side: \[ \frac{2}{5} \times -6 = -\frac{12}{5} \] \[ -\frac{12}{5} - 6 \] Since 6 can be rewritten as \( \frac{30}{5} \), the left side becomes: \[ -\frac{12}{5} - \frac{30}{5} = -\frac{42}{5} \]3. Calculate the right side: \[ \frac{7}{5} \times -6 = -\frac{42}{5} \]4. Compare both sides: \[ -\frac{42}{5} = -\frac{42}{5} \] Since both sides are equal, the solution \( w = -6 \) is verified as correct. Verification ensures that you have solved the equation correctly, confirming that the value found satisfies the original equation.