Problem 15 - Algebra for College Students

Question

Translate each verbal sentence into an equation, using \(x\) as the variable. Then solve the equation. $$ \text { The sum of a number and } 6 \text { is }-31 \text { . Find the number. } $$

Step-by-Step Solution

Answer

The number is -37.

Step 1 - Identify the Verbal Sentence Components

Identify the key components in the given verbal sentence: 'The sum of a number and 6 is -31'. Here, the 'number' is represented by the variable x.

Step 2 - Translate Verbal Sentence to an Equation

Translate the identified components into an equation. The phrase 'The sum of a number and 6 is -31' translates to: \[ x + 6 = -31 \]

Step 3 - Isolate the Variable x

Isolate the variable x by subtracting 6 from both sides of the equation. \[ x + 6 - 6 = -31 - 6 \] Simplifying, we get: \[ x = -37 \]

Step 4 - Verify the Solution

Verify the solution by substituting x back into the original equation. If \( x = -37 \), then: \[ -37 + 6 = -31 \] Since this is true, the solution is correct.

Key Concepts

Verbal Sentence Translation

When solving word problems in algebra, the first step is to translate the verbal sentence into a mathematical equation. This means turning the words into symbols and numbers. For example, in the sentence 'The sum of a number and 6 is -31', we recognize that 'sum' implies addition. We also need to determine the unknown quantity, which in this case we represent by the variable \(x\). Therefore, 'the sum of a number and 6' becomes \(x + 6\). The phrase 'is -31' tells us that the result of the sum is -31. Putting this all together, the verbal sentence translates to the equation \(x + 6 = -31\).

Algebraic Equations

Once we've translated the verbal sentence into an equation, we now have an algebraic equation. An algebraic equation is a statement of equality between two expressions. In our case, \(x + 6 = -31\) states that when 6 is added to the unknown number, the result is -31. Algebraic equations can include variables, numbers, and operation symbols (like plus or minus). They are mathematical representations used to solve for unknown values. Understanding how to set up these equations from word problems is crucial in basic algebra.

Isolating Variables

To solve the equation, we need to isolate the variable. Isolating the variable means getting the unknown \(x\) by itself on one side of the equation. Using the equation \(x + 6 = -31\), we can isolate \(x\) by performing the opposite operation to both sides of the equation. Since 6 is added to \(x\), we subtract 6 from both sides: \(x + 6 - 6 = -31 - 6\). This simplifies to \(x = -37\). Now, we've successfully solved for \(x\). We always perform the same operation on both sides of the equation to maintain balance or equality.

Solution Verification

After solving for the variable, it's important to verify the solution. Verification ensures that the solution is correct. To verify \(x = -37\), substitute \(x\) back into the original equation: \(-37 + 6\). Calculate the sum: \(-37 + 6 = -31\). Since the left side of the equation equals the right side (\(-31\)), the solution is verified as correct. This step is crucial because it confirms that our equation and calculations were correct, giving us confidence in our solution.

Basic Algebra

This entire process falls under the realm of basic algebra, which is the foundation for more advanced mathematics. Basic algebra includes operations with numbers and variables, solving equations, and understanding mathematical relationships. The skills learned here, like translating sentences into equations, isolating variables, and verifying solutions, are fundamental. Mastering basic algebra concepts will help you solve a wide range of problems, both in academics and real life. Practice these steps to build a strong mathematical foundation.