Problem 53 - Prealgebra

Question

Translate each of the following and simplify the result. What number do you subtract from \(-3\) to get \(-9 ?\)

Step-by-Step Solution

Answer

Subtract 6 from -3 to get -9.

Step 1: Identify the Unknown

Let the unknown number be represented by the variable \( x \). We need to find a number that, when subtracted from \(-3\), will result in \(-9\). This scenario can be represented by the equation: \( -3 - x = -9 \).

Step 2: Set Up the Equation

Write down the equation that describes the situation: \( -3 - x = -9 \). This equation logically means that if you take \( x \) from \(-3\), the result should be \(-9\).

Step 3: Isolate the Variable

To find \( x \), we need to isolate it on one side of the equation. Start by adding \( x \) to both sides to get rid of the subtraction of \( x \): \( -3 = -9 + x \).

Step 4: Solve for the Variable

Now, to solve for \( x \), add 9 to both sides of the equation: \( -3 + 9 = x \). This simplifies to \( 6 = x \). So, \( x \) is 6.

Key Concepts

Solving Equations with Variables

In prealgebra, solving equations with variables is a foundation. This skill is all about finding what number, when used in place of a variable, makes the equation true. The variable, often represented by letters like \( x \), is like a placeholder for the unknown number we need to discover.
For example, in the equation \(-3 - x = -9\), the task is to figure out what number \( x \) should be for both sides to equal each other. It's like playing detective with numbers!
Here are some simple steps to solve equations with variables:

• Identify the Variable: What is the unknown number? Use a letter to represent it.
• Set Up the Equation: Write down the situation as an equation. This equation expresses the relationship between known numbers and the unknown variable.
• Isolate the Variable: Use mathematical operations like addition or subtraction to get the variable alone on one side of the equation. This may involve reversing operations on both sides.
• Solve the Equation: Finally, do the math to find the value of the variable.

The beauty of using variables is that they allow you to set up and solve equations for nearly any situation, making them super useful in math and real life.

Integer Subtraction

Subtracting integers, such as negative and positive whole numbers, can be tricky at first, but becomes simple with practice. Integer subtraction involves finding the difference between two integers.
For instance, when dealing with negative numbers, remember that subtracting a negative number is the same as adding its positive counterpart. Visualize it this way:

• Think of negative numbers as steps backward. If you subtract more steps backward, it's like moving forward.
• Flip the Sign: Subtracting a negative is like adding a positive.
• Subtract normally when dealing with positive numbers. Simply count how far one number is from the other.

Applying this to our equation, \(-3 - x = -9\), we realize that solving involves subtracting negative numbers. It’s important to simplify and switch subtraction of a negative to an addition. This might feel strange at first, but it’s all about flipping the idea of moving backward into moving forward!

Solving Linear Equations

Solving linear equations is about finding the value of the variable that makes a linear equation true. Linear equations can be visualized as straight lines when graphed, and solving them means finding the point where the line intersects the horizontal axis (at \( y = 0 \)).
Here's how you can solve a simple linear equation like \(-3 - x = -9\):

• Rearrange the Equation: Start by moving terms around to isolate the variable. You want to get \( x \) by itself. In this case, by adding \( x \) to both sides, you get \(-3 = -9 + x\).
• Perform Arithmetic Operations: Simplify the equation as you move terms. Add 9 to both sides to solve for \( x \), which gives you \(-3 + 9 = x\).
• Final Calculation: Simplify the arithmetic to find the value of \( x \). Here, \(6 = x\), which means the number we were looking for is 6.

Each step carefully maintains the balance of the equation, ensuring that what you do to one side, you do to the other. The joy of solving comes when you’ve isolated \( x \) and uncovered the hidden number. Try practicing this with different numbers, and soon enough, solving linear equations will feel like second nature!