91Ó°ÊÓ

Equations are fundamental in mathematics. They are statements that assert the equality of two expressions. Most commonly, equations are represented with variables, such as \(x\) or \(y\), which stand for unknown values waiting to be uncovered. An equation often contains constants, coefficients, and one or more variables. The main idea is to solve the equation for the unknowns to make the statement true.Consider the equation \(50 - y = 20\). Here, \(y\) is the variable, and our goal is to find its value. We're told to check if \(30\) is a solution for this equation. This simply means that if we substitute \(y\) with \(30\) in the equation, the statement should hold true, i.e., both sides should equal the same value. By understanding the parts of an equation and their roles, we can tackle more complex mathematical challenges efficiently.

The substitution method is a straightforward approach that is often used in algebra to solve equations. It's based on replacing a variable with a specific value to see if it satisfies the equation.

How to Use Substitution

• Identify the variable you need to solve for. In our example, it is \(y\).
• Substitute the given number into the equation in place of the variable. Here, we substitute \(y\) with \(30\), resulting in the equation \(50 - 30 = 20\).
• Perform the arithmetic operation: subtract \(30\) from \(50\), which gives us \(20\).

Through substitution, if both sides of the equation are equal after substituting the variable, this indicates that the solution is correct. For this example, since the result is indeed equal to \(20\), it confirms that our substitution was successful, making \(30\) the solution.

Solution verification is the final step in problem-solving. After substituting a value and performing calculations, it is crucial to ensure the results are valid. It involves checking if both sides of an equation are equal once the proposed solution is inserted.

Verifying the Solution

• Ensure you perform every arithmetic operation correctly. Here, we verified \(50 - 30 = 20\).
• Compare both sides of the equation. If they match, as they did in \(20 = 20\), then the number provided is indeed a solution.
• If they don't match, reconsider your substitution or check the calculations for errors.

This step is vital as it confirms accuracy and completeness in solving mathematical problems. Accurate solution verification wraps up the problem-solving process by providing confidence in the results obtained.