91Ó°ÊÓ

In this section, we'll delve into the concept of equation distribution. When solving linear equations, we often encounter parentheses. To simplify the equation, we need to distribute, or multiply, the term outside the parentheses by each term inside the parentheses. For example, in the equation \( 70 = 10(3t - 2) \), we need to distribute the 10 to both \( 3t \) and \( -2 \). This process involves applying the distributive property of multiplication over addition or subtraction. Distributing the 10 gives us:
\[ 70 = 10 \times 3t - 10 \times 2 \]
This simplifies to:
\[ 70 = 30t - 20 \] Distributing is a crucial first step in solving equations as it helps to eliminate parentheses and simplify the equation, making it easier to work with.

Next, we need to understand isolating variables. After distributing and simplifying the equation, our goal is to get the variable (in this case, \( t \)) on one side by itself. The equation \( 70 = 30t - 20 \) can be solved by isolating the term with the variable. To do this, we add 20 to both sides to cancel out the constant term on the right side. The process looks like this:
\[ 70 + 20 = 30t - 20 + 20 \]
Simplifying, we get:
\[ 90 = 30t \] Now, the term with our variable \( 30t \) is isolated on one side, and all constants are moved to the other side. Isolating the variable is a key step that simplifies the equation to a more manageable form, bringing us closer to finding the solution.

Finally, we solve for the variable. With the variable isolated, we can now solve for it. Our current simplified equation is \( 90 = 30t \). To find the value of \( t \), we need to get \( t \) alone. We do this by performing the inverse operation of multiplication, which is division. We divide both sides by 30:
\[ \frac{90}{30} = \frac{30t}{30} \]
Simplifying this yields:
\[ t = 3 \] By dividing both sides by the coefficient of \( t \) (which is 30), we solve for \( t \). This final step completes our solution, showing that \( t = 3 \). Understanding how to solve for a variable is essential for solving equations, as it yields the value that satisfies the original equation. Happy learning!