Problem 60 - Algebra A Combined Function

Question

The amount \(y\) of land occupied by farms in the United States (in millions of acres) from 1997 through 2007 is given by \(y=-4 x+967\). In the equation, \(x\) represents the number of years after 1997 . (Source: National Agricultural Statistics Service) a. Complete the table. $$ \begin{array}{|c|c|c|c|} \hline \boldsymbol{x} & 4 & 7 & 10 \\ \hline \boldsymbol{y} & & & \\ \hline \end{array} $$ b. Find the year in which there were approximately 930 million acres of land occupied by farms. (Hint: Find \(x\) when \(y=930\) and round to the nearest whole number.) c. Use the given equation to predict when the land occupied by farms might be 900 million acres. (Use the hint for part b.)

Step-by-Step Solution

Answer

a. 951, 939, 927; b. 2006; c. 2014.

Step 1: Understand the equation

The given equation is \( y = -4x + 967 \), where \(y\) represents the land in millions of acres, and \(x\) is the number of years after 1997. We need to find \(y\) for specified values of \(x\), and \(x\) for specified values of \(y\).

Step 2: Fill the table for part a

Calculate \(y\) using the equation \( y = -4x + 967 \) for each \(x\):- For \(x = 4\): \[ y = -4(4) + 967 = -16 + 967 = 951 \]- For \(x = 7\): \[ y = -4(7) + 967 = -28 + 967 = 939 \]- For \(x = 10\): \[ y = -4(10) + 967 = -40 + 967 = 927 \]So the completed table is:| \(x\) | 4 | 7 | 10 ||-------|----|----|----|| \(y\) | 951| 939| 927|

Step 3: Solve for part b

We need to find the year when \(y = 930\). Solve the equation \(930 = -4x + 967\):\[936 = 967 - 930\]\[ -4x = 930 - 967 = -37 \]\[ x = \frac{-37}{-4} = 9.25\]Since \(x = 9.25\), rounding to the nearest whole number gives \(x = 9\). Therefore, the year is 1997 + 9 = 2006.

Step 4: Solve for part c

We need to find when \(y = 900\). Solve the equation \(900 = -4x + 967\):\[900 = -4x + 967\]\[-4x = 900 - 967 = -67\]\[ x = \frac{-67}{-4} \approx 16.75\]Rounding \(16.75\) gives \(x = 17\), so the prediction is for the year 1997 + 17 = 2014.

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. This method allows for the systematic solving of problems. In this exercise, we use algebra to predict future land occupancy by farms.
The fundamental task here is solving the linear equation given in the form of:

• \( y = -4x + 967 \)

This equation is linear because both the variables \(x\) and \(y\) are raised to the power of one.
The coefficients provide us crucial information where \(-4\) is the slope, showing the rate of change, and \(967\) is the y-intercept, the starting point when \(x=0\).
In contextual terms, the equation reflects a decrease by 4 million acres per year, beginning from 967 million acres in 1997. Problems like this reinforce the concept that algebra can turn real-world situations into mathematical representations, to help make informed predictions.

Table Completion

Table completion involves filling out a set of values in a table, usually based on a mathematical rule or pattern. In this exercise, we utilize the given equation \( y = -4x + 967 \) to determine the value of \(y\) for specific values of \(x\).
For instance, substituting \(x = 4\) into the equation, you get \( y = -4(4) + 967 = 951 \). Similar calculations complete the table for other values:

• \( x = 7 \), \( y = -4(7) + 967 = 939 \)
• \( x = 10 \), \( y = -4(10) + 967 = 927 \)

Filling these values into a table helps visualize the trend of decreasing land occupation each year. Understanding and practicing table completion deepens your grasp of how linear equations relate to data points, and prepare you for more complex problem-solving scenarios.

Year Calculation

For year calculation, we convert values of \(x\) derived from the equation into actual years, considering \(x\) is the number of years after a base year, 1997 in this case.
Let's look at an example:

• When calculating when there were approximately 930 million acres, we set \(y = 930\) in the equation.
• By solving, \(930 = -4x + 967\), you find \(x = 9.25\), which indicates 9.25 years after 1997.

Rounding 9.25 gives you \(x = 9\), leading to the year 2006.
This kind of problem trains you on converting mathematical solutions into real-year contexts, allowing practical interpretation and application of abstract solutions.

Prediction

Prediction using linear equations involves using the equation parameters to forecast future events or values. In our example, predicting when the land size reaches 900 million acres is crucial.
To make this prediction:

• Set \(y = 900\) in the equation \(900 = -4x + 967\).
• Solving gives \(x = 16.75\), which upon rounding becomes \(x = 17\).

Adding 17 to the year 1997 predicts that this scenario might occur in 2014.
This method showcases the usefulness of linear equations in forecasting scenarios. It empowers us to make decisions based on probable future outcomes detailed by numerical data. Understanding such predictions strengthens your ability to utilize mathematics in real-world applications.