91Ó°ÊÓ

The substitution method is a straightforward way to solve and check equations. It involves replacing a variable with a given number to see if the equation holds true. For the equation in this problem, we check whether the values of 3 and -3 for the variable \(w\) make both sides of the equation equal.

This process includes:

• Substituting the value of the variable into the equation.
• Calculating the expressions on both sides of the equation.
• Comparing the results to see if they match.

Let's break down the substitution process for the given equation, \(11-5w=-1-w\), with the values \(w=3\) and \(w=-3\):
1. Substitute \(w=3\):
\text{\(11-5(3)=-1-3\)}
This simplifies to \(-4=-4\), which is true.

2. Substitute \(w=-3\):
\text{\(11-5(-3)=-1-(-3)\)}
This simplifies to \(26=2\), which is false.

Using the substitution method helps us easily test if the values provided satisfy the equation.

Simplifying equations is a crucial step in solving algebra problems. It involves reducing expressions to their simplest forms. When we simplify both sides of an equation, it becomes easier to determine whether they are equal. Here’s how it was done for this problem:

For \(w=3\), substitute and simplify:
\text{\(11-5(3)=-1-3\)}
Calculate the left side: \text{\(11-15=-4\)}
Calculate the right side: \text{\(-1-3=-4\)}
Both sides simplify to -4, so the equation holds true.

For \(w=-3\), substitute and simplify:
\text{\(11-5(-3)=-1-(-3)\)}
Calculate the left side: \text{\(11 + 15 = 26\)}
Calculate the right side: \text{\(-1 + 3 = 2\)}
The left side simplifies to 26, and the right side simplifies to 2. Since 26 is not equal to 2, the equation does not hold true for \(w=-3\).

Simplifying equations consistently helps ensure that the steps are correct and makes it easier to see if both sides of an equation are equal.

Verifying your solutions is the final, essential step in solving equations. This process confirms that the values you've found or been given actually satisfy the equation. Here’s a recap on how to verify solutions for the given values:

1. For \(w=3\), we substitute and simplify:\br> \text{\(11-5(3)=-1-3\)} simplifies to \(-4=-4\). Since both sides match, \(w=3\) is a valid solution.

2. For \(w=-3\), we substitute and simplify:
\text{\(11-5(-3)=-1-(-3)\)} simplifies to \(26=2\). Since both sides do not match, \(w=-3\) is not a valid solution.

Verification involves:

• Substituting the given values back into the original equation.
• Simplifying both sides of the equation.
• Ensuring that both sides are equal for the values to be valid solutions.

Always verify your solutions to ensure your calculations and substitutions are correct.