Problem 32 - Algebra A Combined Function

Question

Solve the following. Four times a number added to 5 is divided by 6 . The result is \(\frac{7}{2} .\) Find the number.

Step-by-Step Solution

Answer

The number is 4.

Step 1: Set Up the Equation

Let's start by translating the word problem into a mathematical equation. We have "four times a number added to 5 is divided by 6" and the result of this process is \(\frac{7}{2}\). We can set this up as \(\frac{4x + 5}{6} = \frac{7}{2}\), where \(x\) is the unknown number we need to find.

Step 2: Clear the Fraction

To eliminate the fraction, multiply both sides of the equation by 6. This will give us:\[4x + 5 = \frac{7}{2} \times 6\]Simplifying the right side, we get:\[4x + 5 = 21\]

Step 3: Isolate the x Term

Subtract 5 from both sides to isolate the \(4x\) term:\[4x + 5 - 5 = 21 - 5\]\[4x = 16\]

Step 4: Solve for x

Finally, divide both sides by 4 to solve for \(x\):\[x = \frac{16}{4}\]Thus, \(x = 4\).

Key Concepts

Equations

Equations are mathematical statements where two expressions are equal. They consist of elements like variables, constants, and operators. In this problem, we transformed a word problem into an algebraic equation. The equation is given as \( \frac{4x + 5}{6} = \frac{7}{2} \). Here, \(x\) represents the unknown number. Equations form the basis for solving many algebra problems. By expressing relationships mathematically, equations help us find unknown values using known values. In many algebra problems, setting up the equation is often the first step. Be sure to carefully translate all parts of the word problem into the equation.

Fractions

Fractions represent a part of a whole and are essential in solving algebraic equations. In our equation \( \frac{4x + 5}{6} = \frac{7}{2} \), both sides include fractions. This is common in problems involving proportional distributions or repeated multiplication or division.

• Numerator: This is the top part of the fraction, which represents the number of parts we have.
• Denominator: The bottom part indicates the total number of equal parts the whole is divided into.

In our case, the fraction on the left side of the equation involves \(4x + 5\) as the numerator and 6 as the denominator. Understanding how to handle fractions is crucial in algebra as it aids in simplifying equations, especially when they involve division or require balancing both sides.

Isolate Variable

To isolate a variable means to manipulate the equation to have the variable on one side and everything else on the other. It's a crucial step in determining the value of the unknown. After clearing the fraction and simplifying, our equation became \(4x + 5 = 21\). To isolate \(x\), we:

• Subtract 5 from both sides to remove the constant and focus on the term involving \(x\).
• We were left with \(4x = 16\), which made it easier to solve for \(x\) itself.

This manipulation helps directly solve for the unknown. Isolation is often achieved through addition or subtraction to remove constants, and multiplication or division helps deal with coefficients. Mastering this concept is key to solving equations efficiently.

Multiplication and Division

Multiplication and division are fundamental operations that help manage the coefficients and break down complex parts of equations. In our problem, we began by multiplying both sides by 6 to clear the fraction. This gives us an equivalent equation without fractions, which is much easier to handle. After isolating \(4x\), the next step involved division—the opposite of multiplication. We divided both sides by 4 to solve for \(x\), i.e., \(x = \frac{16}{4} \), resulting in \(x = 4\).

• Multiplication helps in combining smaller units to form a whole, especially when clearing denominators in fractions.
• Division assists in breaking down values and reversing multiplication operations.

Understanding these operations and their interplay in equations is vital. It allows you to manipulate and solve equations systematically. Use them effectively to simplify and solve for the unknown values.