Problem 49 - Elementary Algebra Within Reach

Question

The sum of two numbers \(x\) and \(y\) is 20 and the difference of the two numbers is 2 . The system of equations that represents this situation is $$ \left\\{\begin{array}{l} x+y=20 \\ x-y=2 \end{array}\right. $$ Solve the system graphically to find the two numbers.

Step-by-Step Solution

Answer

The intersection point of the two line graphs represents the solution to the system of equations, which are the values of \(x\) and \(y\).

Step 1: Set up the Equations

From the problem, two equations can be formed: \(x + y = 20\) and \(x - y = 2\).

Step 2: Plot the First Equation

Graph the first equation onto a coordinate plane. It can be done by choosing values for \(x\) and finding the corresponding \(y\) values.

Step 3: Plot the Second Equation

Similarly, plot the second equation onto the same coordinate plane by choosing values for \(x\) and finding the corresponding \(y\) values.

Step 4: Find the Intersection Point

The solution to the system of equations is the point where the lines intersect. Inspect the graph or use mathematical methods to determine the point of intersection.

Key Concepts

Algebraic Equations

Algebraic equations are the backbone of solving mathematical problems involving unknowns. An equation is essentially a statement that two expressions are equal, often featuring variables like \( x \) and \( y \). In the case of our exercise, the two equations \( x + y = 20 \) and \( x - y = 2 \) represent a system that we are trying to solve.

Solving algebraic equations often involves finding the value of the unknown variables that make the equation true. For instance, when we say \( x + y = 20 \), we are looking for all the pairs of numbers \( (x, y) \) that, when added together, give us 20. Similarly, \( x - y = 2 \) seeks pairs where the difference between \( x \) and \( y \) is 2. In a system of equations, we aim to find particular values that satisfy all equations simultaneously.

Algebraic equations can be simple, with a single variable and one solution, or they can be part of a more complex system, like the one in our exercise. In the context of systems of equations, solving them graphically involves representing each equation as a line on a coordinate plane and finding the point or points where they meet.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graph relationships between two variables. It is divided into four quadrants by a horizontal axis (the x-axis) and a vertical axis (the y-axis). Each point on this plane is defined by an ordered pair of numbers \( (x, y) \), which corresponds to horizontal and vertical positions respectively.

To solve equations graphically, as we do with our system of equations, we utilize the coordinate plane to create a visual representation of the equations. For the equation \( x + y = 20 \), we can choose a few values for \( x \) and calculate the corresponding \( y \) values to get points that we can then plot on the plane. We perform the same process for the equation \( x - y = 2 \). Each equation forms a straight line, and where they cross over is of particular interest.

Understanding how to plot points and lines on the coordinate plane is crucial for visually solving systems of equations. The graphical approach not only aids in finding solutions but also helps in understanding the relationship between variables and the nature of their interaction — whether they are directly or inversely related, for example.

Intersection of Lines

The concept of the intersection of lines is fundamental when solving systems of equations graphically. When two lines on the coordinate plane cross each other, they meet at a point known as their point of intersection. This single point is significant because its coordinates represent the solution to the system of equations — the values of \( x \) and \( y \) that satisfy both equations simultaneously.

In our exercise, after graphing the lines represented by \( x + y = 20 \) and \( x - y = 2 \), we look for where these lines intersect. At this point, both equations hold true. This method of finding the solution is particularly useful when it's difficult to solve the system algebraically, or when one wishes to verify the solution obtained by algebraic means.

To locate the intersection accurately, it may be necessary to use more precise graphing tools or algebraic methods if the graph does not provide a clear answer. However, the visual method often gives a quick and intuitive understanding of how many solutions a system might have — one in the case of two intersecting lines, none if they are parallel, or infinitely many if they coincide.