Problem 54 - Algebra for College Students

Question

Solve each equation. $$\frac{6}{a+2}=\frac{a}{a+2}$$

Step-by-Step Solution

Answer

a = 6.

Step 1 - Understand the Equation

Given the equation off:off:off:$$\frac{6}{a+2} = \frac{a}{a+2}$$. The denominators on both sides of the equation are equal: off:Since offoff:noff:\both sides have the same denominator, the numerators must also be equal for the equation to hold true.

Step 2 - Set Numerators Equal

Set the numerators equal to each other: $$6 = a$$.

Step 3 - Solve for a

Since the equation $$6 = a$$ is already solved for $$a$, the solution to the equation is $$a = 6$$.

Key Concepts

Setting Numerators Equal

In solving rational equations, one helpful method is setting the numerators equal when the denominators are the same. Let's understand this better. Given the equation \( \frac{6}{a+2}=\frac{a}{a+2} \), notice that the denominators on both sides are \( a + 2 \). Whenever the denominators of two rational expressions are equal, the numerators must also be equal for the equation to hold true.
This is why we set the numerators equal: \( 6 = a \). This approach simplifies our work, making it easier to solve for the variable involved. Remember: only set the numerators equal if the denominators are exactly the same.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Understanding them is key to solving rational equations.
In our example, \( \frac{6}{a+2} \) and \( \frac{a}{a+2} \) are both rational expressions. The numerator in the first expression is '6' and the denominator is \( a + 2 \). Similarly, for the second expression, the numerator is 'a' and the denominator is \( a + 2 \).
Working with these expressions involves:

• Ensuring the denominators are not zero (since division by zero is undefined).
• Simplifying the expressions when possible.
• Setting the numerators equal if the denominators already match, as we did in the previous section.

Simplification is often a crucial first step in solving equations involving rational expressions.

Solving for a Variable

Once we set the numerators equal, solving for the variable becomes straightforward. In the equation \( 6 = a \), we only need to isolate the variable 'a'.
Here, the equation is already in its simplest form since 'a' is isolated on one side: \( a = 6 \).
General steps for solving for a variable include:

• Isolating the variable on one side of the equation.
• Performing inverse operations to move constants and coefficients. For example, if we had \( 6 - 2 = a \), we would add 2 to both sides to isolate 'a'.
• Simplifying the resulting expression to find the value of the variable.

These steps ensure we accurately determine the variable's value. Always check your solution by substituting it back into the original equation to verify its correctness.