Problem 31 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 3. $$ \frac{2}{3} c=10 $$

Step-by-Step Solution

Answer

The solution is \( c = 15 \).

Step 1: Identify the Equation

The given equation is \( \frac{2}{3} c = 10 \). This is a simple algebraic equation where the variable \( c \) is multiplied by \( \frac{2}{3} \).

Step 2: Clear the Fraction

To isolate \( c \), we need to eliminate the fraction by multiplying both sides of the equation by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \).

Step 3: Perform Multiplication

Multiply both sides by \( \frac{3}{2} \), resulting in:\[ c = 10 \times \frac{3}{2} \]

Step 4: Calculate the Right Side

Calculate \( 10 \times \frac{3}{2} = 15 \). This implies \( c = 15 \).

Step 5: Check the Solution

Substitute \( c = 15 \) back into the original equation for verification:\[ \frac{2}{3} \times 15 = 10 \]Solving this gives \( 10 = 10 \), which confirms our solution is correct.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are the backbone of algebra and essentially contain numbers, variables, and arithmetic operations. They allow us to represent complex relationships using a combination of letters and numbers. In our exercise, the algebraic expression is \( \frac{2}{3}c = 10 \). Here, \( c \) is a variable representing an unknown value, and \( \frac{2}{3} \) is a coefficient that scales \( c \).

When working with algebraic expressions, our goal is often to solve for the unknown variable. This means isolating the variable on one side of the equation. By understanding algebraic expressions, we can manipulate these relationships and find solutions to various problems.

Key components of algebraic expressions include:

• Variables: symbols used to represent unknown values (e.g., \( c \)).
• Coefficients: numbers that multiply the variable (e.g., \( \frac{2}{3} \)).
• Constants: fixed values that don’t change (e.g., 10).

Comprehending these elements will greatly enhance your ability to tackle algebraic problems efficiently.

Grasping Reciprocal Multiplication

Reciprocal multiplication is a powerful technique used to simplify equations that involve fractions. The reciprocal of a number is simply one divided by that number. For example, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).

In our equation \( \frac{2}{3}c = 10 \), to solve for \( c \), we need to get rid of the fraction \( \frac{2}{3} \). We do this by multiplying both sides by its reciprocal. This effectively "cancels out" the fraction, leaving the variable by itself.

Why does this work? When we multiply a number by its reciprocal, the result is always 1. So in the context of a variable, multiplying by the reciprocal lets the variable "stand alone."

• The reciprocal of \( a/b \) is \( b/a \).
• Multiplying by a reciprocal undoes the original fraction operation.
• It is a key step in isolating variables in algebraic equations.

Understanding this concept helps in making equations more manageable and finding solutions quickly.

Handling Fractional Coefficients

Dealing with fractional coefficients is a skill that becomes critical when solving algebraic equations. A fractional coefficient, like \( \frac{2}{3} \) in our equation \( \frac{2}{3}c = 10 \), is a fraction that multiplies a variable. This can initially seem complicated, but by using reciprocal multiplication, we efficiently handle these fractions.

Here are steps to manage fractional coefficients:

• Identify the fractional coefficient in the equation.
• Find the reciprocal of the fractional coefficient.
• Multiply both sides of the equation by this reciprocal to eliminate the fraction.

This approach transforms the equation quickly, making it simpler to solve. Remember, the motive is to make the equation as straightforward as possible by "clearing" fractions. Mastering this process will greatly aid in solving a wide range of algebraic problems. The simplicity in recognizing and managing these fractions is key to progressing in algebra.