91Ó°ÊÓ

When solving linear equations, the goal is to make the variable of interest, in this case, \( t \), stand alone on one side of the equation. This process is known as isolating the variable. In the given equation \(-11 = 5t - 9\), the first step to isolate \( t \) is to eliminate the constant term attached to it, which is \(-9\).
To do this, we add 9 to both sides of the equation. Mathematical problems often require performing the same operation on both sides to maintain equality. By doing so, the equation becomes:

• \( -11 + 9 = 5t - 9 + 9 \)

Simplifying it results in \( -2 = 5t \), effectively isolating \( t \) with the constant removed. Remember, the sequence of operations should aim at gradually freeing \( t \) from any terms that accompany it. This allows you to find its exact value.

After solving an equation, it’s important to verify that the solution is correct. This process is known as checking the solution. Once we found \( t = -\frac{2}{5} \), it's a good practice to replace \( t \) in the original equation and see if both sides remain equal.
Consider our equation:

• \(-11 = 5\left(-\frac{2}{5}\right) - 9\)

Replace \( t \) with \(-\frac{2}{5}\) and simplify using basic arithmetic.

• The left side remains as \(-11\).
• The right side simplifies, as we perform the multiplication and addition, to \(-2 - 9\).

The result is \(-11 = -11\), confirming that the solution is true. It reassures that our calculations for isolating and solving \( t \) were correct, providing confidence in our final answer.

The distributive property is a valuable tool for simplifying expressions and solving equations. It’s particularly helpful when you have an equation that involves both multiplication and subtraction or addition. In our exercise, this property helps in verifying the correctness of the solution.
When we had to check the solution \( t = -\frac{2}{5} \), we performed the following operation:

• Distribute the multiplication over addition: \( 5(-\frac{2}{5}) = -2 \).

The distributive property states that \( a(b + c) = ab + ac \). This property ensures you correctly apply multiplication across all terms within the parentheses.
While solving the initial check, the distributive property guaranteed that each term was accurately handled, leading to the true equality \(-11 = -11\). Mastery of this property greatly simplifies the problem-solving process, especially for more complex equations. Understanding and utilizing properties like these can save time and reduce errors in elementary algebra.