Problem 67 - Prealgebra

Question

What number do you add to -3 to get -7?

Step-by-Step Solution

Answer

Add -4 to -3 to get -7.

Step 1: Identify the Structure of the Problem

The problem is asking for a number that, when added to -3, results in -7. This can be rewritten as an equation: \(-3 + x = -7\).

Step 2: Isolate the Unknown Variable

To find \(x\), we need to isolate it on one side of the equation. We do this by subtracting \(-3\) from both sides of the equation: \(-3 + x - (-3) = -7 - (-3)\).

Step 3: Simplify the Equation

Subtracting \(-3\) is the same as adding 3, so we simplify: \(x = -7 + 3\).

Step 4: Calculate the Result

Perform the calculation: \(-7 + 3 = -4\). Thus, \(x = -4\).

Key Concepts

Solving Linear Equations

Linear equations, particularly those involving a single variable, form the bedrock of algebra. The goal is always to isolate the unknown variable, which in turn reveals its value. In the given exercise, we deal with an equation \( -3 + x = -7 \), where we need to find the value of \( x \). This is a classic example of solving a linear equation in one variable.

To solve, follow these crucial steps:

• Rewrite the problem as an equation if it's expressed in words. Here, "What number do you add to -3 to get -7?" becomes \( -3 + x = -7 \).
• Isolate the variable \( x \) on one side of the equation. This often involves using inverse operations – here, subtracting \(-3\) from itself and from the other side of the equation. We conclude that \( x \) is equivalent to solving: \( -7 + 3 \).
• Calculate the value by performing the arithmetic operation. The result of \( -7 + 3 \) is \( -4 \), indicating \( x = -4 \).

Simple applications like these form a foundation from which more complex algebraic concepts can be built.

Integer Operations

Integer operations are fundamental in all areas of mathematics and prealgebra. They include addition, subtraction, multiplication, and division involving whole numbers and their negatives. Understanding integer operations allows students to effectively handle equations and mathematical expressions.

In the exercise, we dealt specifically with addition of negative numbers, which can sometimes be counterintuitive.

• When adding negative numbers, envision the number line. Essentially, adding \(-3\) implies moving three steps to the left from zero, and adding a negative number again similarly affects movement on this line.
• Subtraction of negatives translates to addition. Therefore, when we shifted from \( -7 + 3 \), thinking of it as reversing direction on the number line helps clarify the operation to reach \(-4\).
• More complex interactions can occur between negative integers and positive figures. Mastering negative and positive number operations solidifies your arithmetic foundation.

Practicing these operations enhances one's comfort in solving algebraic problems.

Prealgebra Concepts

Prealgebra serves as the introductory phase into more structured algebraic thinking. It involves understanding numbers, basic operations, and how to manipulate them according to established rules. These concepts lay the groundwork for solving algebraic equations like the one in the exercise.

Consider these foundational prealgebra ideas:

• **Understanding Equations**: An equation is a statement declaring equality between two expressions. Solving it involves finding the unknown quantity that balances them.
• **Operation Inverses**: Every arithmetic operation has an inverse. Addition versus subtraction or multiplication versus division. To isolate variables, you'll frequently utilize this relationship.
• **Number Properties**: Familiarity with properties like commutivity, associativity, and distributive laws aid in comprehending how numbers behave in operations and equations.

Building upon these prealgebra concepts is vital for progression in mathematics, equipping students to tackle more advanced problems with confidence.