Problem 9 - Fundamentals of Mathematics

Question

If four times a number is increased by fifteen, the result is five. What is the number?

Step-by-Step Solution

Answer

The number is \(-\frac{5}{2}\).

Step 1: Set Up the Equation

Identify the variable for the unknown number. Let's call this unknown number \( x \). According to the problem, four times the number increased by fifteen equals five. This can be represented with the equation: \[ 4x + 15 = 5 \]

Step 2: Isolate the Term with the Variable

To solve for \( x \), we need to isolate the term that contains \( x \). First, subtract 15 from both sides of the equation to eliminate the constant term.\[ 4x + 15 - 15 = 5 - 15 \] Simplifying both sides, we get: \[ 4x = -10 \]

Step 3: Solve for the Variable

Now, divide both sides of the equation by 4 to solve for \( x \).\[ \frac{4x}{4} = \frac{-10}{4} \] Simplifying, we find: \[ x = -\frac{5}{2} \]

Step 4: Verification

To verify the solution, substitute \( x = -\frac{5}{2} \) back into the original equation to check if both sides are equal.\[ 4\left(-\frac{5}{2}\right) + 15 \] Simplify:\[ -10 + 15 = 5 \]. Since both sides equal 5, the solution is verified.

Key Concepts

Problem Solving

When we encounter a problem, especially in algebra, it's essential to take a structured approach to find a solution. Problem-solving usually involves identifying the unknowns, understanding the relationship between the known and unknown parts, and translating words into mathematical expressions.

• Identify the Unknown: Start by figuring out what you need to solve for. In word problems, this usually involves numbers or quantities that have been described in the problem statement.
• Translate Words to Symbols: Convert the verbal problem into a mathematical equation. This includes identifying keywords that imply mathematical operations: 'increased by' often means addition, 'decreased by' suggests subtraction, 'times' indicates multiplication, and 'is equal to' sets the stage for an equation.
• Solve Step-by-Step: Break down the equation into more manageable parts and solve it step-by-step. Being methodical and orderly helps ensure you don't miss any important steps or details.

By following these structured steps, problem-solving becomes focused and less overwhelming.

Linear Equations

Linear equations are the backbone of many algebra problems and are characterized by their constant rates of change. A linear equation is an equation of the first degree, meaning it has no exponents greater than one.

Linear equations typically take the form of \( ax + b = c \) where:

• \( x \) is the variable
• \( a \) is the coefficient of the variable
• \( b \) is the constant term
• \( c \) is the sum or result

A key property of linear equations is that they graph as straight lines. Solving a linear equation often involves manipulating it to isolate the variable, leading us to the solution rather straightforwardly. The equation in our example, \( 4x + 15 = 5 \), is a classic linear equation.

Variable Isolation

Variable isolation is a crucial skill in solving algebraic equations. The goal is to have the variable on one side of the equation and numbers on the other. This simplifies finding the value of the variable.

Here are general steps for isolating the variable:

• Simplify Each Side: Combine like terms on both sides if necessary.
• Remove Add/Subtract Terms: Use addition or subtraction to move constants to the opposite side of the equation. For example, in solving \( 4x + 15 = 5 \), subtracting 15 from both sides gives \( 4x = -10 \).
• Clear Coefficients: Divide or multiply to remove any coefficients next to the variable. For \( 4x = -10 \), divide both sides by 4 to get \( x = -\frac{5}{2} \).

Consistently applying these steps will lead to solving the equation efficiently. Remember, each manipulation is driven by keeping the equation balanced: whatever is done to one side must be done to the other.