Problem 28 - Prealgebra

Question

Solve each equation. Check your solution. $$-21=\frac{k}{8}$$

Step-by-Step Solution

Answer

The solution is \(k = -168\).

Step 1: Understand the Equation

The equation given is \[-21 = \frac{k}{8}\]. This means that \(k\) divided by \(8\) is equal to \(-21\). We need to find the value of \(k\).

Step 2: Isolate the Variable

To find \(k\), we must get rid of the division by \(8\). We can do this by multiplying both sides of the equation by \(8\):\[-21 \times 8 = \frac{k}{8} \times 8\].

Step 3: Compute the Multiplication

Calculating \(-21 \times 8\) gives us:\[-168 = k\].So, \(k = -168\).

Step 4: Check the Solution

Replace \(k\) with \(-168\) in the original equation to verify:\[\frac{-168}{8} = -21\].After calculating, \(-168 \div 8\) indeed equals \(-21\), confirming our solution is correct.

Key Concepts

Solving Equations

Solving equations is like solving puzzles. You're trying to find the unknown piece, which is the variable, to complete the picture. An equation, such as \[-21 = \frac{k}{8}\], tells us two things are equal. The left side is known, while the right side includes the mystery variable. Our job is to make both sides match by discovering the value of this variable.

• Think of the equation as a balance scale. Whatever you do to one side, you must do to the other to keep it balanced.
• The goal in solving is to isolate the variable on one side of the equation, showing what it equals.
• Each step in solving involves performing operations like addition, subtraction, multiplication, or division to simplify the equation.

Understanding the goal helps us move towards finding the value of the unknown variable.

Integer Multiplication

When it comes to integer multiplication, we are multiplying whole numbers or their negatives. In the equation \(-21 = \frac{k}{8}\), we need to undo the division for isolation of the variable. Here, we used integer multiplication to handle this.

• Multiplying an equation by a number is like multiplying both weights on a balance scale by the same amount, so it stays balanced.
• Negative integers, like -21, multiply the same way as positive integers, but the result is negative if multiplied with a positive number.

Since our equation involved -21, multiplying both sides by 8 helps cancel out the division by 8 of the variable, thus, 8 times -21 gives us -168. This step ensures that we now have \(k = -168\).

Division of Integers

Dividing with integers involves splitting a number into equal parts. In our equation, the variable was divided by 8. The operation \(\frac{k}{8}\) represents this division. To reverse this action, we need to perform the opposite calculation.

• Positive divided by positive gives positive, negative divided by positive gives negative, and similarly for other sign combinations.
• Division is like distributing something equally among a set number of people or items; in this case, the value for the variable was split into 8 parts.

Thus, to solve \(\frac{k}{8} = -21\), the solution involved reversing the division by multiplying by 8, allowing us to isolate the variable on one side.

Variable Isolation

Isolating the variable is a key step in solving any equation. This process involves moving everything except the variable to the other side of the equation. Start by performing inverse operations to get the variable alone.

• Look at what operation is affecting the variable. If it’s division, like in our equation, use multiplication as the inverse operation to isolate it.
• Always apply these operations equally to both sides to maintain balance in the equation.
• Recheck your steps to ensure you haven't altered any values incorrectly during the process.

By multiplying both sides by 8, we eliminated the division from the variable's side, getting \(k\) alone: Originally: \(-21 = \frac{k}{8}\), After isolation: \(k = -168\).It confirms the exact value for \(k\), solving the equation completely.