Detailed Analysis of Precipitation Graphs and Algebraic Linear Equations

Evaluation of Initial Numerical Data and Answer Choices

The provided material begins with a list of discrete numerical values and alphabetical options that likely correspond to a preceding set of questions not fully captured in the excerpt. These include the characters and values $Q$, $50$, and $00$. Additionally, a set of multiple-choice answers is presented: option B) $157$, option C) $314$, and option D) $7,854$. These figures are followed by the formal start of Question 11.

Comparative Analysis of Monthly Precipitation Trends

Question 11 involves the interpretation of a line graph that monitors environmental data within a specific geographic area designated as a valley. The purpose of this graph is to track monthly precipitation levels over a defined period of $6$ months. In this statistical context, the valley's recorded figures are plotted to show fluctuations in rainfall or snowfall over time.

The specific objective of this question is to determine between which two consecutive months the valley experienced the greatest decrease in recorded precipitation. A decrease is represented on a line graph by a downward slope, where the value at the end of the interval is lower than the value at the beginning. The "greatest" decrease refers to the interval with the largest negative change or the steepest downward trajectory. The consecutive intervals provided for analysis are:

• A) Month $1$ to Month $2$

• B) Month $2$ to Month $3$

• C) Month $3$ to Month $4$

• D) Month $4$ to Month $5$

Algebraic Methodology for Solving Linear Equations

Question 12 presents a mathematical challenge requiring the isolation of an unknown variable, $x$. The linear equation provided is as follows: $2(3x + 5) = 4x + 16$

To effectively solve for $x$, one must adhere to the standard order of operations and algebraic principles. The first step involves removing the parentheses by applying the distributive property. This means the coefficient of $2$ must be multiplied by every term inside the parentheses: $2 \times 3x = 6x$ $2 \times 5 = 10$

This distribution results in the simplified expression on the left side of the equation: $6x + 10 = 4x + 16$

Sequential Calculation to Determine the Variable Value

Once the parentheses are removed, the next step is to consolidate like terms. This is achieved by moving all terms containing the variable $x$ to one side of the equation and all constant numbers to the opposite side. Subtracting $4x$ from both sides of the equation yields: $6x - 4x + 10 = 16$ $2x + 10 = 16$

Following this, the constant term $10$ is subtracted from both sides to isolate the term containing the variable: $2x = 16 - 10$ $2x = 6$

Finally, the value of the variable $x$ is isolated by dividing both sides of the equation by the coefficient $2$. The final calculation is: $x = \frac{6}{2}$ $x = 3$

Through this multi-step logical process—distribution, combination of like terms, isolation of the variable, and division—the value of $x$ is confirmed to be $3$.