Sharon And John's Ages Solving A System Of Equations Graphically
In the realm of mathematical puzzles, age-related problems often present intriguing challenges. These problems require us to translate verbal information into algebraic equations and then solve them using various techniques. One such problem involves Sharon and John, whose ages are intertwined in a fascinating way. The sum of Sharon's and John's ages is 70, and Sharon is four times as old as John. Our task is to unravel this age enigma by representing it as a system of equations and exploring its graphical representation. This article delves into the steps involved in formulating the equations, graphing them, and interpreting the solution within the context of the problem.
Defining the Variables and Equations
To embark on our mathematical journey, we first need to define the variables that will represent the unknowns. Let's denote Sharon's age by the variable 's' and John's age by the variable 'j'. With these variables in place, we can now translate the given information into algebraic equations.
The first piece of information states that the sum of Sharon's and John's ages is 70. This translates directly into the equation:
s + j = 70
This equation represents a linear relationship between Sharon's and John's ages. It signifies that the combined age of the two individuals is a constant value.
The second piece of information reveals that Sharon is four times as old as John. This can be expressed as the equation:
s = 4j
This equation establishes a direct proportionality between Sharon's and John's ages. It indicates that Sharon's age is always four times that of John's.
Together, these two equations form a system of linear equations:
s + j = 70
s = 4j
This system of equations encapsulates the age relationship between Sharon and John. To solve this system, we can employ various methods, including substitution, elimination, or graphing. In this article, we will focus on the graphical method to visualize the solution.
Graphing the System of Equations
To graph the system of equations, we need to represent each equation as a line on a coordinate plane. The x-axis will represent John's age (j), and the y-axis will represent Sharon's age (s). To graph each line, we need to find two points that lie on the line. This can be done by substituting values for one variable and solving for the other.
Graphing the Equation s + j = 70
Let's start by graphing the equation s + j = 70. To find two points on this line, we can substitute values for j and solve for s.
- When j = 0:
This gives us the point (0, 70).s + 0 = 70 s = 70 - When j = 70:
This gives us the point (70, 0).s + 70 = 70 s = 0
Plotting these two points on the coordinate plane and drawing a line through them gives us the graph of the equation s + j = 70. This line represents all possible combinations of Sharon's and John's ages that add up to 70.
Graphing the Equation s = 4j
Next, let's graph the equation s = 4j. Again, we need to find two points on this line.
- When j = 0:
This gives us the point (0, 0).s = 4 * 0 s = 0 - When j = 10:
This gives us the point (10, 40).s = 4 * 10 s = 40
Plotting these two points and drawing a line through them gives us the graph of the equation s = 4j. This line represents all possible combinations of Sharon's and John's ages where Sharon's age is four times that of John's.
Finding the Solution Graphically
The solution to the system of equations is the point where the two lines intersect. This point represents the values of s and j that satisfy both equations simultaneously. By visually inspecting the graph, we can identify the point of intersection.
The point of intersection appears to be at approximately (14, 56). This means that John's age (j) is approximately 14 years, and Sharon's age (s) is approximately 56 years. To verify this solution, we can substitute these values back into the original equations.
- s + j = 70
56 + 14 = 70 70 = 70 (True) - s = 4j
56 = 4 * 14 56 = 56 (True)
Since both equations are satisfied, the graphical solution (14, 56) is indeed the correct solution to the system of equations. Therefore, John is 14 years old, and Sharon is 56 years old.
Alternative Methods for Solving the System
While we have solved this problem graphically, it's worth noting that alternative methods, such as substitution and elimination, can also be used to find the solution. These methods involve algebraic manipulation of the equations to isolate one variable and solve for it. Then, the value of that variable is substituted back into the other equation to find the value of the remaining variable. These methods provide an alternative approach to solving systems of equations and can be particularly useful when graphical solutions are not easily obtainable.
Importance of Solving Systems of Equations
The ability to solve systems of equations is a fundamental skill in mathematics and has wide-ranging applications in various fields. Systems of equations arise in numerous real-world scenarios, such as economics, engineering, physics, and computer science. They are used to model and solve problems involving multiple variables and constraints. For instance, in economics, systems of equations can be used to determine the equilibrium prices and quantities in a market. In engineering, they can be used to analyze the forces and stresses in structures. In physics, they can be used to model the motion of objects. Mastering the techniques for solving systems of equations empowers individuals to tackle complex problems and make informed decisions in a variety of contexts.
Conclusion
In this exploration, we have successfully navigated the age puzzle involving Sharon and John. By translating the given information into a system of equations and graphing them, we were able to visualize the solution and determine their ages. John is 14 years old, and Sharon is 56 years old. This problem serves as a testament to the power of mathematics in unraveling real-world scenarios. Furthermore, we discussed the importance of solving systems of equations and highlighted their diverse applications in various fields. As we conclude this mathematical journey, let us appreciate the elegance and practicality of mathematics in solving problems and making sense of the world around us.
This problem exemplifies the power of algebra in representing real-world situations. By using variables and equations, we can transform a word problem into a mathematical model that can be solved using graphical or algebraic techniques. Understanding how to set up and solve systems of equations is a crucial skill in mathematics and has applications in various fields.
Understanding how the equations translate into lines on a graph is also important. Each linear equation represents a line, and the solution to the system is the point where the lines intersect. This graphical representation provides a visual understanding of the problem and the solution.
This type of problem can also be solved algebraically using substitution or elimination methods. Exploring these different solution methods can enhance your problem-solving skills and provide a deeper understanding of the underlying concepts.
Problem Statement and Equations
The core of the problem lies in translating the word problem into a mathematical representation. We are given two crucial pieces of information:
-
The sum of Sharon's age (s) and John's age (j) is 70: This can be written as the equation:
s + j = 70 -
Sharon is 4 times as old as John: This translates to the equation:
s = 4j
These two equations form a system of linear equations, which we can solve to find the individual ages of Sharon and John. The question asks us to identify the graph that represents this system, but before we look at the graphical representation, let's delve deeper into the meaning of these equations and how they relate to each other.
Understanding the Equations
The first equation, s + j = 70, represents a linear relationship between Sharon's age and John's age. It tells us that for every year John gets older, Sharon's age decreases by one year to maintain the sum of 70. This equation can be visualized as a straight line with a negative slope on a graph where the x-axis represents John's age (j) and the y-axis represents Sharon's age (s).
The second equation, s = 4j, indicates a direct proportionality between Sharon's age and John's age. It states that Sharon's age is always four times John's age. This equation also represents a straight line, but in this case, the line passes through the origin (0,0) and has a positive slope of 4. This means that as John's age increases, Sharon's age increases four times as fast.
Solving the System Algebraically
Before we consider the graphical representation, let's solve the system of equations algebraically. This will give us the exact ages of Sharon and John and help us verify the graphical solution later. We can use the substitution method since the second equation already has s isolated.
Substitute s = 4j into the first equation:
4j + j = 70
Combine like terms:
5j = 70
Divide both sides by 5:
j = 14
So, John's age is 14 years. Now, substitute j = 14 back into the equation s = 4j:
s = 4 * 14
s = 56
Therefore, Sharon's age is 56 years. We have found that John is 14 years old and Sharon is 56 years old. This solution satisfies both equations in the system.
Graphical Representation of the System
The graphical representation of a system of equations provides a visual way to understand the solution. Each equation represents a line on a coordinate plane, and the solution to the system is the point where the lines intersect. This intersection point represents the values of the variables that satisfy both equations simultaneously.
Plotting the Lines
To plot the lines, we need to consider the slope and y-intercept of each equation. Let's rewrite the first equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:
s + j = 70
s = -j + 70
In this form, we can see that the slope is -1 and the y-intercept is 70. This means the line starts at the point (0, 70) on the y-axis and goes down one unit for every one unit it moves to the right.
The second equation, s = 4j, is already in slope-intercept form, with a slope of 4 and a y-intercept of 0. This line starts at the origin (0, 0) and goes up four units for every one unit it moves to the right.
Identifying the Intersection Point
The point where the two lines intersect represents the solution to the system of equations. Visually, this point is where the line representing s + j = 70 and the line representing s = 4j cross each other on the graph. Based on our algebraic solution, we know that the intersection point should be (14, 56), where j = 14 and s = 56.
Finding the Correct Graph
The correct graph of the system of equations will show two lines intersecting at the point (14, 56). One line will have a negative slope and a y-intercept of 70, representing the equation s + j = 70. The other line will have a positive slope of 4 and pass through the origin, representing the equation s = 4j.
Analyzing Different Graphical Representations
When presented with multiple graph options, it's crucial to analyze each graph carefully. Look for the following features:
- Slopes of the lines: Are the slopes positive or negative, and do they match the equations?
- Y-intercepts: Do the lines cross the y-axis at the correct points?
- Intersection point: Do the lines intersect, and if so, is the intersection point at (14, 56)?
By carefully considering these features, you can identify the graph that accurately represents the system of equations.
Conclusion
In summary, the problem of Sharon's and John's ages highlights the power of using systems of equations to solve real-world problems. By translating the word problem into algebraic equations and then representing those equations graphically, we can find the solution visually and algebraically. The key is to understand the relationships between the variables, formulate the equations correctly, and interpret the graphical representation accurately. This approach is not only applicable to age problems but also to a wide range of mathematical and scientific challenges.
This example underscores the importance of being able to translate word problems into mathematical equations, and conversely, being able to interpret graphical representations of mathematical relationships. The ability to solve systems of equations, whether algebraically or graphically, is a fundamental skill in mathematics and has numerous applications in various fields. The combination of algebraic manipulation and graphical visualization provides a powerful tool for problem-solving and a deeper understanding of mathematical concepts.
