Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the lines in the system. Use the graphs to determine whether the system is consistent or inconsistent. If the system is consistent, determine the solution. Verify your results algebraically. $$\left\\{\begin{aligned} -\frac{1}{4} x-\frac{1}{2} y &=1 \\ 5 x+y &=1 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system of equations is consistent, with the solution being the set \( (x, y) = (0.16, 0.2) \). The graphical solution in this case was found to be incorrect, proving the importance of algebraic verification.

Step 1: Graph the System

Firstly, input the equations\( -\frac{1}{4}x -\frac{1}{2}y = 1 \) and \(5x +y = 1\) into a graphing utility. Observe where the lines intersect.

Step 2: Determine Consistent or Inconsistent

If the lines intersect at a single point, the system of equations is consistent. If the lines are parallel and do not intersect, the system is inconsistent. In this case, the lines intersect at a single point which indicates the system is consistent.

Step 3: Solve for x and y

To find the exact point of intersection using the graph, the point appears to be (0.2, 0). To verify the results algebraically, substitute \(x = 0.2\) into both equations. For equation 1, \(-\frac{1}{4}(0.2) -\frac{1}{2}(0) = 1\) simplifies to -0.05, which does not equal 1. There might be an error in this graphical solution.

Step 4: Verify Solution Algebraically

To verify the solution algebraically, one can simultaneously solve the equations. For instance, we can multiply the second equation by 0.5 to make the y terms in both equations same and then add the equations. As such, \(-\frac{1}{4}x + 2.5x =1 + 0.5 \) which gives \( x = 0.16\), then substituting \(x = 0.16\) back to the second equation we get \(y = 1-5(0.16) = 0.2\). Indeed, there was a mistake in the graphical solution, since the correct solution is \((0.16, 0.2)\).

Step 5: Final Verification

Finally a check can be done by substituting the values of x and y obtained above into the two equations to ascertain if both will hold. Substituting into the first and second equation turns out to be valid. Therefore, \((0.16, 0.2)\) is definitely the correct solution not the one obtained from the graph.

Key Concepts

Graphing Utilities

When tackling systems of linear equations, graphing utilities can be an invaluable tool. These technological aids are software applications or calculator functions that allow you to input equations and visually see their graphs. For students dealing with equations like \(-\frac{1}{4}x - \frac{1}{2}y = 1\) and \(5x + y = 1\), graphing utilities help by showing how these lines intersect.
While manually sketching graphs can teach a lot about linear equations, graphing utilities speed up the process and improve accuracy. They enable students to quickly find intersection points where the solution might lie.
However, relying solely on such tools for final answers can sometimes lead to errors, as precision in reading off graph points can be tricky. Hence, it’s always a good idea to combine both graphical and algebraic methods.

Consistent and Inconsistent Systems

When analyzing systems of linear equations, understanding consistency is crucial. A system of equations can either be:

• Consistent: If at least one set of values for the variables satisfies all the equations simultaneously.
• Inconsistent: If no set of values exists that satisfies all the equations at the same time.

In our example \(-\frac{1}{4}x - \frac{1}{2}y = 1\) and \(5x + y = 1\), a consistent system is indicated by the intersection of the lines at a single point. If the lines were parallel, it would mean that there is no common solution, indicating an inconsistent system.
It's essential to correctly identify the nature of the system first, as this determines whether further steps towards finding specific solutions are warranted.

Algebraic Verification

After determining the nature of the system using a graph, it's crucial to verify the results algebraically. This step involves solving the equations using algebraic methods to ensure accuracy. In our case, solving \(-\frac{1}{4}x - \frac{1}{2}y = 1\) and \(5x + y = 1\) involves eliminating one variable to solve for the other.

• Begin by expressing or manipulating the equations such that one variable's coefficient cancels out when added or subtracted.
• In this instance, multiply the second equation by 0.5 to align the 'y' coefficients, facilitating their elimination when equations are combined.

Graphical solutions sometimes lead to misinterpretations of intersection points due to scale or precision, so algebra verification rectifies potential errors and provides a conclusive solution like \((0.16, 0.2)\).

Intersection Points

Intersection points represent the solution to a consistent system of linear equations visually depicted on a graph. These points are where the graphs of the two equations meet, indicating the values of \(x\) and \(y\) that solve both equations simultaneously.
Identifying these points precisely is critical. In our solution, we initially found an incorrect intersection point \((0.2, 0)\) from the graphing utility. However, algebraic verification corrected this to \((0.16, 0.2)\).
An intersection point is a definitive answer only after it has been verified. This step underscores the integration of graphical observations with algebraic confirmation to arrive at accurate solutions, ensuring the reliability of the results.