Problem 65 - Introductory Algebra for College Students

Question

In use the given information to write an equation. Let \(x\) represent the number described in each exercise. Then solve the equation and find the number. Negative three times a number, increased by \(15,\) is \(-6\)

Step-by-Step Solution

Answer

The number is 7.

Step 1: Formulate the equation

Given statement is 'Negative three times a number, increased by 15, is -6'. This translates into the equation \(-3x + 15 = -6\) where \(x\) is the unknown number we're solving for.

Step 2: Isolate the variable term

Subtract 15 from both sides of the equation to isolate the \(-3x\) term, resulting in \(-3x = -6 - 15\). This simplifies to \(-3x = -21\).

Step 3: Solve for the variable

Divide both sides of the equation by -3. The solution to the equation is \(x = -21/-3\), which simplifies to \(x = 7\).

Key Concepts

Understanding Problem-Solving in Algebra

Algebra is like a puzzle, and solving algebraic equations is about finding the missing piece. To tackle a problem effectively, we first need to understand what is being asked. In the given exercise, we were asked to find a number such that when it is multiplied by negative three and increased by 15, the result is -6. The goal is to create an equation that accurately represents the problem.

• Identify what you are solving for. Here, it's the number expressed as \(x\).
• Translate words into a mathematical equation. "Negative three times a number, increased by 15, is -6" becomes \(-3x + 15 = -6\).
• Solve the equation by systematically unraveling it to find the value of \(x\).

By setting up the equation correctly, we turn the word problem into a solvable mathematical equation. Problem-solving in algebra usually involves these steps: understanding the problem, setting up the equation, and then solving for the variable.

Mastering Variable Isolation

Variable isolation is crucial in solving algebraic equations. The objective is to get the variable \(x\) by itself on one side of the equation. This allows you to see its value clearly.

• Begin by looking at your equation, \(-3x + 15 = -6\).
• The first step is to eliminate any constants on the side of the equation with the variable. Here, subtract 15 from both sides to remove the constant. This gives you \(-3x = -21\).
• Now focus on isolating \(x\) by getting rid of the coefficient attached to it. Since \(-3x\) means \(-3\) times \(x\), you would divide both sides by \(-3\) to isolate the variable, resulting in \(x = 7\).

By treating both sides of the equation equally, through operations like addition, subtraction, multiplication, or division, we gradually isolate \(x\). This method ensures that we maintain the equality and eventually uncover the value of the variable.

Handling Negative Numbers

Negative numbers can be tricky, especially when they appear in equations. However, with a few rules, you can handle them with ease. Let's break down how we handled negative numbers in our exercise.

• When translating the problem into an equation, watch for signs. "Negative three times a number" translates to \(-3x\).
• Always follow arithmetic rules with negative values during operations. For instance, we stopped at \(-3x = -21\). To solve, we divide both sides by \(-3\). Remember, dividing two negatives yields a positive: \(-21 \div -3 = 7\).
• Avoid common pitfalls: \(-(-a)\) becomes \(a\), and \(-a - b\) equals \(-(a + b)\).

With practice, negative numbers become less confusing. Always double-check your signs at each step to ensure accuracy, as handling negatives correctly is vital to achieving the right solution.