- Express the system of equations: \(l - e = 4\) and \(l + e = 20\).
- Convert to slope-intercept form: \(e = l - 4\) and \(e = -l + 20\).
- Solve the system by adding the equations to get \(2l = 24\), yielding \(l = 12\), then substitute to find \(e = 8\).
- Conclude that Lian is 12 years old and Ella is 8 years old, which corresponds to the point \((12, 8)\) on a graph.
- The final answer is \(\boxed{l=12, e=8}\).
Explanation
- Problem Analysis
We are given the ages of Lian (\(l\)) and her sister Ella (\(e\)). We know that they are 4 years apart, meaning \(l - e = 4\), and the sum of their ages is 20, meaning \(l + e = 20\). We are asked to model this situation using a system of linear equations and solve it to find their ages. The system of equations is:
\[ \begin{cases} l - e = 4 \ l + e = 20 \end{cases} \]
Here, \(l\) represents Lian's age and \(e\) represents Ella's age.
- Slope-Intercept Form Conversion
To solve this system, we can express each equation in the slope-intercept form \(e = ml + b\), where \(e\) is the dependent variable (y-axis) and \(l\) is the independent variable (x-axis).
For the first equation \(l - e = 4\), subtracting \(l\) from both sides gives \(-e = -l + 4\), which simplifies to \(e = l - 4\). This line has a slope of 1 and an \(e\)-intercept of \((0, -4)\).
For the second equation \(l + e = 20\), subtracting \(l\) from both sides gives \(e = -l + 20\). This line has a slope of -1 and an \(e\)-intercept of \((0, 20)\).
- Solving the System of Equations
We can solve the system by adding the two original equations:
\[(l - e) + (l + e) = 4 + 20\]
\[2l = 24\]
\[l = 12\]
Substituting \(l = 12\) back into the first equation \(l - e = 4\):
\[12 - e = 4\]
\[e = 8\]
Thus, Lian is 12 years old and Ella is 8 years old. These values represent the intersection point \((12, 8)\) of the two lines on a coordinate plane.
- Graphing and Conclusion
The two linear equations represent lines on a coordinate plane.
The line \(e = l - 4\) passes through points \((4, 0)\) and \((12, 8)\).
The line \(e = -l + 20\) passes through points \((20, 0)\) and \((12, 8)\).
The intersection of these two lines is at \((12, 8)\), which provides the solution to the ages of Lian and Ella. Therefore, Lian is 12 years old and Ella is 8 years old. The final answer is \(\boxed{l=12, e=8}\).
Examples
Understanding systems of linear equations is crucial for balancing budgets or managing resources. For instance, if you have a fixed total amount of money to spend on two items and a specific required price difference between them, you can set up a system of linear equations to determine the exact price of each item. In our case, the sum of ages and their difference allows us to pinpoint their specific ages by finding the point where the constraints intersect on a graph.