Explain the mistake that is made. Solve the equation (4)/(p)-3=(2)/(5 p). Solution: Cross multip

step 1 In this problem, what we're being asked to do is to find the mistake from the given student's wo

Explain the mistake that is made. Solve the equation...

Explain the mistake that is made.
Solve the equation $\frac{4}{p}-3=\frac{2}{5 p}$.
Solution:
Cross multiply.
$$
\begin{aligned}
(p-3) 2 &=4(5 p) \\
2 p-6 &=20 p \\
-6 &=18 p \\
p &=-\frac{6}{18} \\
p &=-\frac{1}{3}
\end{aligned}
$$
This is incorrect. What mistake was made?

Key Concepts

Algebraic Manipulation and Distribution

Algebraic manipulation involves applying operations such as distribution, combining like terms, and moving terms from one side of the equation to the other. It is crucial that any multiplication or distribution is applied correctly across all relevant terms in the equation. A common mistake is to incorrectly distribute multiplication over terms, which can fundamentally alter the structure of the equation and lead to an incorrect solution.

Rational Equations

Rational equations are equations in which the variables appear in the denominators of fractions. Solving these equations typically requires careful handling of the expressions, such as finding a common denominator to combine terms or clearing the fractions by multiplying by the least common denominator. This ensures that the manipulation of the equation following the removal of denominators is done correctly without introducing extraneous mistakes.

Cross Multiplication Technique

Cross multiplication is a method used to solve equations that involve proportions, by multiplying the numerator of one fraction by the denominator of the other and vice versa. However, this technique must be applied correctly by ensuring that the equation is set up as a proper proportion. Inappropriate or incomplete cross multiplication, such as only partially multiplying or misidentifying the terms, can lead to errors in the subsequent algebraic steps.

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