Answer :
To explain to your boss the solution to the system of equations for the number of imports and exports based on the given data, we start by understanding the data in the table:
[tex]\[ \begin{array}{|l|l|l|} \hline \text{Month} & f(x)= \text{No. of imports} & g(x)= \text{No. of exports} \\ \hline \text{January} (1) & 3 & 3 \\ \hline \text{February} (2) & 6 & 4 \\ \hline \text{March} (3) & 9 & 5 \\ \hline \text{April} (4) & 12 & 6 \\ \hline \end{array} \][/tex]
From the table, we want to determine the point at which the number of imports equals the number of exports.
### Step-by-Step Solution:
1. Formulate Linear Equations:
- We can start by finding the equations of the lines that represent imports [tex]\( f(x) \)[/tex] and exports [tex]\( g(x) \)[/tex].
2. Finding the Equation for Imports [tex]\( f(x) \)[/tex]:
The data points for imports are: [tex]\( (1, 3) \)[/tex], [tex]\( (2, 6) \)[/tex], [tex]\( (3, 9) \)[/tex], [tex]\( (4, 12) \)[/tex].
Observing these, we notice that for each increase of 1 month, the imports increase by 3. This suggests a linear relationship. Therefore,
[tex]\[ f(x) = 3x + 0 \][/tex]
Here, the slope (rate of change) is 3, and the y-intercept is 0.
3. Finding the Equation for Exports [tex]\( g(x) \)[/tex]:
The data points for exports are: [tex]\( (1, 3) \)[/tex], [tex]\( (2, 4) \)[/tex], [tex]\( (3, 5) \)[/tex], [tex]\( (4, 6) \)[/tex].
Similarly, for each increase of 1 month, the exports increase by 1, indicating a linear relationship. Therefore,
[tex]\[ g(x) = 1x + 2 \][/tex]
Here, the slope is 1, and the y-intercept is 2.
4. Solving the System for Intersection:
To find when the imports equal the exports, set [tex]\( f(x) \)[/tex] equal to [tex]\( g(x) \)[/tex]:
[tex]\[ 3x = 1x + 2 \][/tex]
Subtract [tex]\( 1x \)[/tex] from both sides:
[tex]\[ 2x = 2 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 1 \][/tex]
5. Interpret the Solution:
The solution [tex]\( x = 1 \)[/tex] signifies that the imports and exports are equal in the first month (January). At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 3(1) = 3 \][/tex]
[tex]\[ g(1) = 1(1) + 2 = 3 \][/tex]
### Summary:
The intersection of the two lines occurs at [tex]\( x = 1 \)[/tex], which means that the number of imports and exports are equal in the month of January. Specifically, 3 imports and 3 exports. This shows that the functions for imports and exports intersect at one point in the given timeframe indicating that only in January, the imports and exports were the same.
This detailed analysis provides a clear mathematical explanation of the given data and their intersection, which can now be clearly communicated to your boss as requested.
[tex]\[ \begin{array}{|l|l|l|} \hline \text{Month} & f(x)= \text{No. of imports} & g(x)= \text{No. of exports} \\ \hline \text{January} (1) & 3 & 3 \\ \hline \text{February} (2) & 6 & 4 \\ \hline \text{March} (3) & 9 & 5 \\ \hline \text{April} (4) & 12 & 6 \\ \hline \end{array} \][/tex]
From the table, we want to determine the point at which the number of imports equals the number of exports.
### Step-by-Step Solution:
1. Formulate Linear Equations:
- We can start by finding the equations of the lines that represent imports [tex]\( f(x) \)[/tex] and exports [tex]\( g(x) \)[/tex].
2. Finding the Equation for Imports [tex]\( f(x) \)[/tex]:
The data points for imports are: [tex]\( (1, 3) \)[/tex], [tex]\( (2, 6) \)[/tex], [tex]\( (3, 9) \)[/tex], [tex]\( (4, 12) \)[/tex].
Observing these, we notice that for each increase of 1 month, the imports increase by 3. This suggests a linear relationship. Therefore,
[tex]\[ f(x) = 3x + 0 \][/tex]
Here, the slope (rate of change) is 3, and the y-intercept is 0.
3. Finding the Equation for Exports [tex]\( g(x) \)[/tex]:
The data points for exports are: [tex]\( (1, 3) \)[/tex], [tex]\( (2, 4) \)[/tex], [tex]\( (3, 5) \)[/tex], [tex]\( (4, 6) \)[/tex].
Similarly, for each increase of 1 month, the exports increase by 1, indicating a linear relationship. Therefore,
[tex]\[ g(x) = 1x + 2 \][/tex]
Here, the slope is 1, and the y-intercept is 2.
4. Solving the System for Intersection:
To find when the imports equal the exports, set [tex]\( f(x) \)[/tex] equal to [tex]\( g(x) \)[/tex]:
[tex]\[ 3x = 1x + 2 \][/tex]
Subtract [tex]\( 1x \)[/tex] from both sides:
[tex]\[ 2x = 2 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 1 \][/tex]
5. Interpret the Solution:
The solution [tex]\( x = 1 \)[/tex] signifies that the imports and exports are equal in the first month (January). At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 3(1) = 3 \][/tex]
[tex]\[ g(1) = 1(1) + 2 = 3 \][/tex]
### Summary:
The intersection of the two lines occurs at [tex]\( x = 1 \)[/tex], which means that the number of imports and exports are equal in the month of January. Specifically, 3 imports and 3 exports. This shows that the functions for imports and exports intersect at one point in the given timeframe indicating that only in January, the imports and exports were the same.
This detailed analysis provides a clear mathematical explanation of the given data and their intersection, which can now be clearly communicated to your boss as requested.
