Problem 14 - Algebra 1: Concepts and Skills

Question

Solve the equation by cross multiplying. Check your solutions. \(\frac{4}{x}=\frac{12}{5(x+2)}\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = -5\).

Step 1: Cross Multiply

Cross multiply \(\frac{4}{x} = \frac{12}{5(x+2)}\) to remove the fractions. This will give \(4*5*(x+2) = 12*x\), which simplifies to \(20x + 40 = 12x\).

Step 2: Form a Quadratic Equation

Subtract \(12x\) from both sides to form a quadratic equation: \(20x - 12x + 40 = 0\), simplifying this will give \(8x + 40 = 0\).

Step 3: Solve the Quadratic Equation

Solve the equation for \(x\). First, subtract 40 from both sides to isolate \(x\): \(8x = -40\). Divide both sides by 8: \(x = -40 / 8\). Thus, \(x = -5\).

Step 4: Check the Solution

Substitute -5 for \(x\) in the original equation to verify the solution: \(\frac{4}{-5} = \frac{12}{5(-5+2)}\). The left side simplifies to \(-0.8\) and the right becomes \(-0.8\), confirming that the solution checks out.

Key Concepts

Solving Equations with Cross Multiplication

Cross multiplication is a method used to eliminate fractions in equations, making them easier to solve. In this problem, we start with the equation \( \frac{4}{x} = \frac{12}{5(x+2)} \). To cross multiply, take the numerator of each fraction and multiply it by the denominator of the other fraction: \( 4 \times 5(x+2) = 12 \times x \).
This process gives us \( 20(x+2) = 12x \). By expanding the left side, we simplify our equation to \( 20x + 40 = 12x \). Cross multiplication is particularly helpful because it removes the fractions, giving us a linear equation that is easier to manipulate and solve.
Remember that cross multiplication only works when two fractions are set equal to each other, and it can simplify many equation-solving problems significantly.

Checking Solutions to Equations

Once we solve an equation, it's crucial to check the solution back in the original equation to ensure it's correct. For our solution \( x = -5 \), we substitute it back into the original equation: \( \frac{4}{-5} = \frac{12}{5(-5+2)} \).
By evaluating each side separately, the left side becomes \(-0.8\). For the right side, simplify \( -5+2 \) to \(-3\) giving \( \frac{12}{5(-3)} \) or \(-0.8\).
This confirms that both sides are equal, proving that \( x = -5 \) is indeed a valid solution. Verification like this is essential because it assures us that no errors were made during calculations, such as division by zero or incorrect algebraic manipulations.

Understanding Quadratic Equations in Context

In the process of solving, we initially seemed to be on track to forming a quadratic equation, but upon simplifying, the equation turned out to be linear. A true quadratic equation takes the form \( ax^2 + bx + c = 0 \), involving the term \( x^2 \).
In our example, after cross multiplying, the equation simplifies directly to \( 8x + 40 = 0 \). Although initially it might seem like a quadratic due to the subtraction stage \( 20x - 12x + 40 \), this simply forms a linear equation since there's no \( x^2 \) term.
The confusion sometimes arises when simplifying expressions intended to be quadratic, only to realize that they're actually linear. Always double-check the presence of \( x^2 \) to distinguish between quadratic and linear equations, as they require different approaches to solve.