Problem 50 - Algebra A Combined Function

Question

Fill in each blank. $$ 20=-4 \cdot $$ ___________

Step-by-Step Solution

Answer

The blank is filled with \(-5\).

Step 1: Understanding the Problem

We are given the equation \( 20 = -4 \cdot x \) and need to fill in the blank with a value of \( x \) that makes the equation true.

Step 2: Isolate the Variable

We need to solve for \( x \) by isolating it on one side. Since \( x \) is currently multiplied by \(-4\), we will divide both sides of the equation by \(-4\) to isolate \( x \).

Step 3: Performing the Division

Divide both sides of the equation by \(-4\):\[ x = \frac{20}{-4} \] This simplifies to \( x = -5 \).

Step 4: Solution Verification

Verify the solution by substituting \( x = -5 \) back into the original equation: \( 20 = -4 \cdot -5 \). Calculate \( -4 \cdot -5 = 20 \), which matches the given equation.

Key Concepts

Equation Solving Steps

When attempting to solve an equation, having a structured approach can make the process straightforward. Here are common steps you can follow:

• Identify the Equation: Locate and clearly state the equation you need to solve. In our example, it was \( 20 = -4 \cdot x \).
• Plan Your Method: Consider how you can rearrange the equation to find the unknown variable. Decide which mathematical operations are needed.
• Isolate the Variable: Perform the necessary steps to get the unknown alone on one side of the equation (more on this later).
• Solve the Equation: Complete the mathematical operations required to determine the value of the unknown.
• Verify Your Solution: Test your solution by substituting it back into the original equation to ensure correctness.

By following these steps, you'll have a roadmap for solving similar equations efficiently. Every step plays a crucial role in reaching the correct solution.

Isolation of Variables

Isolating the variable is an essential step in solving equations that involves positioning the unknown variable all by itself on one side of the equation. This often makes it easier to see what the variable equals. Here's how you can do it effectively:

• Identify the Variable: Look for the unknown in your equation, labeled as \( x \) in our case.
• Perform the Opposite Operation: If the variable term involves multiplication, divide both sides of the equation by that number to isolate it. In the example \( 20 = -4 \cdot x \), we divided both sides by \(-4\).
• Keep Balance: Always perform the same operation on both sides of the equation to maintain equality. This means if you add, subtract, multiply, or divide something on one side, do the same to the other side.

By following these strategies, you'll adeptly shift the variable to one side, paving the way for easy solving.

Integer Division

Integer division comes into play frequently in equation solving, especially when dividing whole numbers. It's essential to understand how it works and how it simplifies equations:

• Definition: Integer division refers to dividing one integer by another, resulting in either a whole number or a fraction reduced to its simplest form.
• Steps: Divide the integers just as you'd do with whole numbers, ignoring any remainders for simplicity if the division isn't exact. Example: \( \frac{20}{-4} = -5 \).
• Simplification: Regular division tells you how one number "fits into" another. When the result is negative, it's important to remember that a positive divided by a negative gives a negative outcome.

Knowing integer division aids tremendously, as it ensures you perform accurate calculations, leading to correct solutions.

Solution Verification

After calculating a solution, it's critical to verify its accuracy. Solution verification ensures the value you found indeed satisfies the original equation:

• Substitute Back: Take the solution you've calculated, substitute it back into the original equation, replacing the variable.
• Calculate: Perform the arithmetic to see if both sides of the equation are equal with your solution in place. In our example, substituting \( x = -5 \) into \( 20 = -4 \cdot x \) results in \( 20 = 20 \).
• Check Consistency: Ensure all operations align with the rules of arithmetic, such as the signs of numbers and the operation rules.

Verification gives you confidence in your answer and is a crucial final step that prevents errors in longer problem-solving sequences.