Problem 32 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Use multiplication to solve the equation. $$ \frac{t}{4}=-4 $$

Step-by-Step Solution

Answer

\(t = -16\)

Step 1: Identification of operation

The first step is to identify the operation that has been performed on the variable \(t\). In this case, \(t\) has been divided by 4. To solve for \(t\), we need to perform the inverse of the operation to both sides of the equation.

Step 2: Applying Inverse Operation

The inverse operation of division is multiplication. Therefore, we multiply both sides of the equation by 4.

Step 3: Solving the Equation

Applying the multiplication gives us \(4* \frac{t}{4} = 4 * -4\), which simplifies to \(t = -16\).

Key Concepts

Solving Equations

When solving equations, our primary goal is to find the value of the unknown variable that makes the equation true. Understanding and solving equations is a fundamental skill in algebra that involves manipulating equations until the variable stands alone on one side. To solve an equation, especially one involving inverse operations, you need to work systematically. You look to eliminate any operations affecting the variable by performing opposite operations, a process which we will discuss in the next section.

• Identify the operation affecting the variable.
• Use the opposite (or inverse) operation to isolate the variable.
• Perform the same operation to both sides of the equation to keep it balanced.

By following these steps, the equation's balance is maintained, leading to the solution that corresponds to the unknown variable.

Inverse Operations

Inverse operations are key in solving equations as they help in canceling out the operations applied to a variable. An inverse operation is the opposite action of a given mathematical operation. If a variable is added to a number, you subtract that number to undo the operation. If a variable is divided by a number, you multiply it by the same number to reverse the division.
For example:

• The inverse of addition is subtraction.
• The inverse of subtraction is addition.
• The inverse of multiplication is division.
• The inverse of division is multiplication.

In our example, since the variable was divided by 4, the inverse operation of multiplying by 4 was applied. This effectively cancels out the division, allowing you to solve for the variable.

Multiplication and Division in Equations

Working with equations often involves multiplication and division, particularly when these operations are applied to variables. When you encounter multiplication or division, you use inverse operations to solve for the variable by isolating it. In our specific exercise, the variable was divided by 4. To solve for the variable (\(t\)), we needed to multiply both sides of the equation by 4.

Here's the process:

• Identify that \(t\) is divided by 4, as in \(\frac{t}{4} = -4\).
• Multiply both sides by 4 to cancel out the division: \(4*\frac{t}{4} = 4*(-4)\).
• This results in \(t = -16\).

By performing these operations equally on both sides, you maintain the equation's balance while isolating the variable to find its value. Multiplication and division make strong tools in your algebra toolkit, especially when paired with understanding inverse operations.