Problem 69 - Prealgebra

Question

What number do you add to -4 to get 3?

Step-by-Step Solution

Answer

Add 7 to -4 to get 3.

Step 1: Understand the Problem

We need to find a number that, when added to -4, results in 3. Symbolically, we can express this as an equation: \(-4 + x = 3\). Our task is to solve for \(x\).

Step 2: Isolate the Variable

To solve for \(x\), we need to isolate it on one side of the equation. We can do this by adding 4 to both sides of the equation: \(-4 + x + 4 = 3 + 4\).

Step 3: Simplify the Equation

Simplify both sides of the equation: The left side becomes \(x\), and the right side becomes 7. Thus, \(x = 7\).

Key Concepts

Utilizing Addition to Solve Equations

Addition is a fundamental operation in mathematics that involves combining two numbers to get a total sum. In equation solving, addition is often used to balance equations and maintain equality. For instance, in the exercise where we are finding a number that, when added to -4, results in 3, addition helps us modify the equation for simplicity. To start, we examine this expression: \(-4 + x = 3\). Here, addition serves as a method to transform one side of the equation without altering the overall balance. Instead of achieving a direct result, we offer a strategic addition of 4 to both sides of the equation:

• This balances the equation by bringing the -4 to zero, leaving just \(x\) on the left side.
• On the right side, adding 4 increases the number 3 to 7.

By using addition strategically, we simplify the equation to easily find the unknown variable.

Understanding Negative Numbers

Negative numbers are numbers less than zero and are vital in various mathematical contexts. Working with negative numbers might seem confusing at first, especially when combined with operations like addition. A negative number essentially represents a deficit or a position to the left of zero on the number line. Let's take the scenario of the exercise, which involves the equation \(-4 + x = 3\). Here, \(-4\) indicates a position 4 units below zero. By adding a positive number to a negative number, you are effectively moving along the number line towards the right side or towards a higher value. This movement continues until all negative units are canceled out, illustrated perfectly when adding 4 to \(-4\):

• \(-4 + 4 = 0\). Once it’s zero, any extra positive units will start counting from zero upwards.
• This shows why when we add 4 to both sides of the equation, we eliminate the negative component completely from one side.

Through understanding negatives, you grasp how operations bring balance to equations.

Isolation of Variable in Equations

The isolation of a variable is a crucial step in solving equations. It refers to the process of manipulating the equation in a way that results in one side containing the variable alone, allowing for an easy determination of its value. In the given exercise where you solve \(-4 + x = 3\) for \(x\), we achieve this isolation by dealing with the \(-4\).

• To isolate \(x\), we add 4 to both sides of the equation, effectively neutralizing the -4.
• Once \(-4 + 4 = 0\), we are left with just \(x\) on the left side of the equation.

This results in the equation \(x = 7\), making it straightforward to see what \(x\) equates to. Isolation is the key to solving for an unknown, allowing you to maintain and understand the balance in equations and ultimately find the exact value of the variables in question.