91Ó°ÊÓ

Solving linear equations is a foundational skill in mathematics. The goal is to find the value of the variable that makes the equation true. In our example, we started with the equation \(\frac{2}{3}x = 24\). To solve for \(x\), we need to isolate the variable on one side of the equation. This often involves performing operations such as addition, subtraction, multiplication, or division.
To isolate \(x\) from \(\frac{2}{3}x = 24\), we needed to eliminate the coefficient \(\frac{2}{3}\) by doing the opposite operation, which in this case involves multiplication by the reciprocal of \(\frac{2}{3}\). Once \(x\) is isolated, you are left with the solution \(x = 36\).
Understanding how to manipulate the equation while maintaining equality on both sides is crucial. It ensures that the integrity of the equation is preserved, and accurate results are achieved.

When solving equations that involve fractions, knowing how to multiply by reciprocals is especially helpful. A reciprocal of a fraction is essentially what you multiply the fraction by to get 1. For example, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\). This is because \(\frac{2}{3} \times \frac{3}{2} = 1\).
In the context of solving equations, multiplying each side of the equation by the reciprocal of the fraction's coefficient is a key step. It helps to cancel out the fraction, making it easier to solve for the variable. In our problem, multiplying both sides of \(\frac{2}{3}x = 24\) by \(\frac{3}{2}\) efficiently isolated \(x\), simplifying it to \(x = 36\).
When performing this operation, it is important to apply the reciprocal multiplication equally to both sides of the equation in order to maintain the equality.

Verification is an important step to ensure the solution you've found is correct. To verify, substitute the solution back into the original equations to see if they equal true statements. In our example, after solving \(\frac{2}{3}x = 24\) and finding \(x = 36\), we checked this against the second equation \(x = 16\).
Substituting, we see that \(36 eq 16\), meaning the solution of the first equation doesn't satisfy the second. This tells us that the two equations are not equivalent because they do not have the same solution.
Always verify both sides after solving because it’s possible to make errors in calculation or in the logical steps, and verification ensures accuracy and confidence in your results.