Problem 63 - Prealgebra

Question

Translate each sentence below into an equation, then solve the equation. The difference of \(5 x\) and 6 is \(-9\)

Step-by-Step Solution

Answer

The solution to the equation is \( x = -\frac{3}{5} \).

Step 1: Understand the Sentence

The given sentence is asking for the difference between 5 times a number (which we'll call \(x\)) and 6, and states that this difference is equal to -9. The phrase "difference of 5 \(x\) and 6" translates to the expression \(5x - 6\).

Step 2: Translate into an Equation

Translate the sentence into a mathematical equation by setting up the expression that represents the difference equal to -9. Thus, the equation becomes: \(5x - 6 = -9\).

Step 3: Solve the Equation

To solve the equation \(5x - 6 = -9\), first add 6 to both sides to isolate the term with \(x\) on one side. This gives:\[ 5x - 6 + 6 = -9 + 6 \]Simplifying, we get:\[ 5x = -3 \]

Step 4: Solve for x

Now, divide both sides of the equation by 5 to solve for \(x\):\[ x = \frac{-3}{5} \]

Step 5: Verify the Solution

Check the solution by substituting \( x = -\frac{3}{5} \) back into the original expression for the difference: \[ 5 \left(-\frac{3}{5}\right) - 6 = -3 - 6 = -9 \]The left side simplifies to -9, verifying that our solution is correct.

Key Concepts

Equation Translation

Equation translation is a crucial skill in math. It involves converting word problems or sentences into mathematical equations. This process helps us understand and solve problems. Imagine you have a sentence that describes a mathematical relationship using everyday language. Your job is to identify key words and phrases, and express them in equation form.

Some common keywords include:

• "Sum" indicates addition.
• "Difference" indicates subtraction.
• "Product" refers to multiplication.
• "Quotient" means division.

In the original exercise, the phrase "the difference of 5 times a number and 6 is -9" translates to the equation \(5x - 6 = -9\). We use the term "difference" to set up a subtraction equation. Translating this correctly forms the foundation for solving the problem.

Prealgebra

Prealgebra is a branch of mathematics that prepares students for algebra. It focuses on concepts necessary to understand and work with algebraic expressions and equations. In prealgebra, you develop skills in understanding basic operations like addition, subtraction, multiplication, and division, and how they apply to unknowns or variables like \(x\).

These skills are essential because they form the basis for your future studies in mathematics. For instance, when handling the equation \(5x - 6 = -9\) from the exercise, prealgebra teaches us how to manipulate this expression to isolate the variable \(x\). This means performing operations such as adding 6 to both sides to simplify the equation. Mastering prealgebra ensures you're ready for more complex algebraic concepts.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations. They represent a specific value or relationship. In the equation translation exercise, expressions help us model the relationship described in the sentence.

For example, the expression \(5x - 6\) consists of:

• "5x" which means 5 times an unknown number (x).
• "-6" which represents a subtraction of 6.

To solve an expression like \(5x - 6\), you need to isolate the variable \(x\). Start by removing constant terms on one side and then simplifying by performing operations. In this exercise, that's done by adding 6 to both sides of the equation, then dividing by 5. Understanding expressions is essential because they are everywhere in mathematics, and mastering them strengthens problem-solving skills.