Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{r}2.5 x-3 y=1.5 \\ x-1.2 y=-3.6\end{array}\right.\)

Step-by-Step Solution

Answer

The system of equations has no solutions. This can be confirmed graphically where the lines representing these equations do not intersect.

Step 1: Multiply the equations to make coefficients of \(y\) cancel each other

Multiply the first equation by 1.2 and the second by 3 to make the coefficient of \(y\) same in both equations. The system then becomes: \[\begin{array}{r}3x-3.6y=1.8 \ 3x-3.6y=-10.8\end{array}\]

Step 2: Eliminate one variable

If we subtract the second equation from the first (or vice versa), variable \(y\) will be eliminated, so we end up with: \[0x + 0y = 12.6\] which implies that 0 = 12.6. This is not possible, so the system of equations has no solution.

Step 3: Graph the system of equations

To confirm our solution, we can graphically represent the system of equations, plotting two straight lines that represent the equations. If the lines do not intersect, this confirms there are indeed no solutions, as intersection points represent common solutions.

Key Concepts

System of Equations

A system of equations is a set of two or more equations that share the same variables. These equations often describe relationships between variables and we aim to find values that satisfy all equations simultaneously. Here is how they typically work:

• Each equation describes a line, plane, or curve depending on the variables involved.
• In most scenarios, you are looking to find the point where these lines or surfaces intersect, indicating a solution that satisfies all involved equations.

In our exercise, the system consists of two linear equations with two variables, x and y. Linear systems like these are commonly solved using methods such as substitution, elimination, or graphing. We use them to find points of intersection, showcasing the potential solutions for the given problem.

Graphing Utility

A graphing utility is a tool, often a calculator or software like Desmos or GeoGebra, that helps visualize equations by plotting their graphs. It serves as a practical way to solve or verify solutions of systems of equations. Here's why it is useful:

• Graphing allows you to see where the graphs of each equation intersect, providing a visual confirmation of solutions.
• It is especially helpful in verifying solutions derived analytically to ensure accuracy.

In the exercise, graphing was used to double-check the system of equations for points of intersection. When plotted, if the lines do not meet, it visually confirms that the system is inconsistent or has no solution. Thus, a graphing utility helps in understanding the behavior and relationships depicted by the equations.

Inconsistent System

An inconsistent system refers to a set of equations that has no solution. This occurs when the lines that represent the equations are parallel, having the same slope but different y-intercepts, meaning they will never meet.

• This can be identified algebraically when simplifying equations leads to a contradiction, such as the statement 0 = 12.6 in our example.
• Graphically, you'll observe that the equations' lines are parallel, confirming there is no common point that satisfies both.

In the given exercise, when we manipulated the equations algebraically, they resolved to the impossible equation 0 = 12.6, hence confirming that the system is inconsistent. No value of x or y could satisfy both equations simultaneously.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols to solve equations and uncover unknown values. It's essential for addressing problems involving systems of equations.

• Algebraic techniques, like elimination, are used to simplify equations to a point where finding solutions is straightforward. The elimination method involves adding or subtracting equations to remove one variable, as seen in the exercise.
• Once a system is reduced, solving becomes a matter of simple arithmetic, or helps in recognizing inconsistencies, as demonstrated by the contradiction 0 = 12.6.

In this exercise, algebra helped us set up and simplify a system of equations by employing the elimination method, illustrating the elegance of this discipline in tackling mathematical challenges.