91Ó°ÊÓ

Understanding how to solve equations is a fundamental skill in algebra. To solve an equation, we aim to find the value of the variable that makes the equation true. In other words, we need to isolate the variable on one side of the equation. This requires using various operations to manipulate the equation while maintaining its balance.

The core idea revolves around performing the same operations on both sides of the equation. Common operations include addition, subtraction, multiplication, and division. Always remember that what you do to one side of the equation, you must do to the other side to keep it balanced. This strategy helps simplify the equation step by step until you solve for the variable.

The multiplicative inverse is a key concept when working with equations involving multiplication. It is also known as the reciprocal.

For a number \(\frac{a}{b}\), its multiplicative inverse is \(\frac{b}{a}\). When you multiply a number by its inverse, the result is always 1.

For example, the multiplicative inverse of \(-\frac{2}{3}\) is \(-\frac{3}{2}\). This is because: \[-\frac{2}{3} \times -\frac{3}{2} = 1\]

In solving equations, we use the multiplicative inverse to isolate the variable. When a variable has a coefficient, multiplying it by the reciprocal of that coefficient will effectively 'cancel out' the coefficient, leaving just the variable.

Once you solve an equation, it's crucial to verify that your solution is correct. This process involves substituting the solution back into the original equation to see if it holds true.

For the given example, the solution found is y = 15.9. To verify this, substitute y = 15.9 back into the original equation:

\(-\frac{2}{3} \times 15.9 = -10.6\)

Then, calculate the left-hand side:
\[-\frac{2}{3} \times 15.9 = -10.6\]

Thus, the left-hand side equals the right-hand side, confirming that y = 15.9 is indeed the correct solution. Verifying your solution helps ensure that no mistakes were made during the solving process, reinforcing the accuracy of your work.