Problem 66 - Algebra A Combined Function

Question

The quotient of a number and \(5,\) minus \(1,\) equals \(\frac{7}{5}\). Find the number.

Step-by-Step Solution

Answer

The number is 12.

Step 1: Set Up the Equation

The problem states that the quotient of a number and 5, minus 1, equals \( \frac{7}{5} \). Let the unknown number be \( x \). We can create the equation: \( \frac{x}{5} - 1 = \frac{7}{5} \).

Step 2: Isolate the Fraction

To solve for \( x \), we first need to get rid of the \(-1\) by adding 1 to both sides of the equation: \( \frac{x}{5} - 1 + 1 = \frac{7}{5} + 1 \). This simplifies to \( \frac{x}{5} = \frac{7}{5} + \frac{5}{5} \).

Step 3: Combine the Fractions

Add the fractions on the right side: \( \frac{7}{5} + \frac{5}{5} = \frac{12}{5} \). Now the equation is \( \frac{x}{5} = \frac{12}{5} \).

Step 4: Solve for the Number

To find \( x \), multiply both sides by 5, cancelling out the denominator: \( x = \frac{12}{5} \times 5 \). Simplifying, we get \( x = 12 \).

Step 5: Verify the Solution

Plug \( x = 12 \) back into the original equation to verify: \( \frac{12}{5} - 1 = \frac{7}{5} \). Simplifying, \( \frac{12}{5} - \frac{5}{5} = \frac{7}{5} \), which confirms our solution is correct.

Key Concepts

Solving Equations

Solving equations is like finding the missing piece of a puzzle. In this exercise, we aim to find the unknown number, represented by the variable \(x\). An equation is a mathematical statement that shows that two expressions are equal, using an equal sign.
For instance, in the equation \(\frac{x}{5} - 1 = \frac{7}{5}\), the left side represents the original problem condition and the right side is the given result.
The process involves applying operations step-by-step to both sides of the equation to isolate and solve for the unknown variable.

• First, we simplify the equation using basic operations such as addition or subtraction. This balances both sides.
• Next, we use multiplication or division to isolate the variable.
• Finally, we verify the solution by plugging it back into the original equation, ensuring correctness.

Approach each step carefully, paying attention to how operations affect the equation's balance, crucial for maintaining equality.

Fractions

Fractions are a way to represent parts of a whole and are often encountered in algebra problems. Understanding fractions is key to solving equations involving them.
In our example, we encounter fractions several times: \(\frac{7}{5}\), \(\frac{5}{5}\), and \(\frac{12}{5}\). These fractions illustrate parts of division results or how expressions are used together to form whole values.
Here’s what you'll frequently do when handling fractions:

• Identify the common denominators, which allow for the addition or subtraction of fractions on the same side of the equation.
• Combine fractions by summing up their numerators and keeping the denominator constant when they have an identical denominator.
• Simplify fractions into whole numbers or simpler fractions when possible.

Grasping these concepts helps simplify problems involving fractions and makes it easier to clear them for solving equations.

Variable Isolation

Variable isolation is about getting the variable on one side of the equation all by itself. It is a crucial skill in solving algebraic equations, and here's how it works in this example:
Initially, we had the equation \(\frac{x}{5} - 1 = \frac{7}{5}\). Our aim was to make \(x\) stand alone.
The steps for isolating a variable usually include:

• Remove any constants or numbers attached to the variable using inverse operations. For example, we added 1 to both sides to cancel out the \(-1\) next to \(\frac{x}{5}\).
• Eliminate fractions by multiplying by denominators, as we multiplied by 5 to resolve \(\frac{x}{5}\) into just \(x\).
• Maintain balance by doing the same operation to both sides of the equation.

Once isolated, solving for the variable becomes straightforward, as seen with \(x = 12\). Mastering variable isolation can greatly assist in more complex algebraic operations, ensuring that you effectively unpack and solve any equation.