Problem 80 - Elementary Algebra

Question

Translate the following sentences into linear equations and then solve. Three-fourths of a number \(x\) is \(9 .\)

Step-by-Step Solution

Answer

The number is 12.

Step 1: Define the equation

The sentence "Three-fourths of a number \(x\) is 9" can be translated into a linear equation by identifying the mathematical components. Here, "Three-fourths of a number \(x\)" can be expressed as \(\frac{3}{4}x\). The sentence then becomes \(\frac{3}{4}x = 9\).

Step 2: Solve for \(x\)

To find \(x\), we need to isolate it on one side of the equation. We do this by multiplying both sides by the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\). So, multiply both sides of the equation:\[ \frac{4}{3} \cdot \frac{3}{4}x = \frac{4}{3} \cdot 9 \]This simplifies to:\[ x = 12 \]

Key Concepts

Solving Linear Equations

Linear equations are mathematical expressions involving variables that are of the first degree, meaning the variable is not raised to any power other than one. They often look like this: \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants.

To solve a linear equation, the goal is to isolate the variable (usually represented by \( x \)) on one side of the equation. This involves using inverse operations to undo any addition, subtraction, multiplication, or division that is affecting the variable. For instance, if you have 3 multiplied by \( x \), you will divide both sides by 3 to solve for \( x \).

In the example, we start with the equation \( \frac{3}{4}x = 9 \). To isolate \( x \), multiply both sides by \( \frac{4}{3} \), the reciprocal of \( \frac{3}{4} \). This allows you to "cancel out" the fraction, making it much easier to find \( x \).

Translating Word Problems

Converting word problems into mathematical equations is a crucial skill in algebra. It helps in identifying the relationship between different elements described in the problem.

To translate a word problem into a linear equation, look for keywords and phrases that indicate mathematical operations:

• "Sum of" suggests addition.
• "Difference" implies subtraction.
• "Product of" indicates multiplication.
• "Quotient of" means division.

In our exercise, "Three-fourths of a number \( x \) is 9" directly translates to \( \frac{3}{4}x = 9 \). Recognizing these clues enables you to form the equation you'll solve to find the unknown quantity.

Reciprocal of a Fraction

The reciprocal of a fraction is achieved by flipping its numerator and denominator. For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).

Reciprocals are particularly useful when solving equations that involve fractions. Multiplying a fraction by its reciprocal results in 1, which effectively "cancels out" the fraction, leaving just the variable or number you want to isolate.

In our original problem, using the reciprocal of \( \frac{3}{4} \) simplifies solving the equation \( \frac{3}{4}x = 9 \). By multiplying both sides of this equation by \( \frac{4}{3} \), we eliminate the fraction and easily determine \( x = 12 \). This step is essential to get rid of fractions in equations and make the solution straightforward.