Problem 82 - Elementary Algebra

Question

Solve each of the following equations. Don't forget that division by zero is undefined. $$\frac{8}{2 x+1}=\frac{4}{x-3}$$

Step-by-Step Solution

Answer

No solution for the equation.

Step 1: Understand the Problem

We need to find the value of \( x \) for the equation \( \frac{8}{2x+1} = \frac{4}{x-3} \). Remember that division by zero is undefined, so ensure that the denominators \( 2x + 1 \) and \( x - 3 \) are not zero.

Step 2: Set the Denominators Not Equal to Zero

To avoid division by zero, set \( 2x+1 eq 0 \) and \( x-3 eq 0 \). Solving these inequalities, we find that \( x eq -\frac{1}{2} \) and \( x eq 3 \). Therefore, these values are excluded from the solution.

Step 3: Cross-Multiply to Eliminate Fractions

Cross-multiply to eliminate the fractions: \( 8(x - 3) = 4(2x + 1) \). This step simplifies the problem to a linear equation without fractions.

Step 4: Distribute and Collect Like Terms

Distribute both sides: \( 8x - 24 = 8x + 4 \). Simplify this to \( 8x - 8x - 24 = 4 \), or \( -24 = 4 \).

Step 5: Analyze the Result

Notice that \( -24 = 4 \) is a contradiction, indicating that no values of \( x \) satisfy this equation subject to the domain restrictions. Therefore, the equation has no solution.

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to solve equations that involve fractions. When you have an equation like \( \frac{a}{b} = \frac{c}{d} \), cross multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa. This gives you a new equation without fractions: \( a \cdot d = b \cdot c \). This method is particularly useful in simplifying problems by eliminating the fractions altogether.In our original exercise, we have \( \frac{8}{2x+1} = \frac{4}{x-3} \). By cross multiplying, the new equation becomes \( 8(x - 3) = 4(2x + 1) \). This transformation simplifies solving, as you can then use algebraic techniques without the complication of fractional terms.When using cross multiplication, always remember to check for any restrictions on the variable, as the original fractions' denominators must not equal zero. Also, be aware of any potential extraneous solutions that could arise from the process.

Division By Zero

Division by zero occurs when a number is divided by zero, and it is important to remember that this operation is undefined in mathematics. Why is this the case? Because dividing by zero would imply that you are distributing a number into an infinite number of parts, which doesn't provide a meaningful result.In our linear equation \( \frac{8}{2x+1} = \frac{4}{x-3} \), it is critical to ensure the denominators are not zero to avoid undefined expressions. To do this, we set the denominators not equaling zero:

• For \( 2x + 1 \), solve \( 2x + 1 eq 0 \), yielding \( x eq -\frac{1}{2} \).
• For \( x - 3 \), solve \( x - 3 eq 0 \), yielding \( x eq 3 \).

These steps ensure our equation only makes sense for values of \( x \) that don't make any denominator zero, which is crucial when solving equations involving fractions.

Solving Equations

Solving equations is all about finding the values of variables that satisfy the equation. Usually, this involves simplifying the equation until the variable is isolated. For linear equations, methods such as substitution, elimination, and manipulation like cross multiplication help achieve this goal.In the problem \( 8(x - 3) = 4(2x + 1) \), we use distribution to expand both sides:

• The left side: \( 8x - 24 \).
• The right side: \( 8x + 4 \).

On simplifying both sides, notice that the equation \( 8x - 8x - 24 = 4 \) leads to \( -24 = 4 \). This statement is false and represents a contradiction. Hence, we conclude that there is no value for \( x \) that will satisfy this equation, given the restrictions previously identified. This situation illustrates how not every equation involving cross multiplication and variable isolation will have a valid solution.