Problem 63 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to find the point(s) of intersection of the graphs. Then confirm your solution algebraically. $$\left\\{\begin{array}{l}4 x^{2}-y^{2}-32 x-2 y=-59 \\ 2 x+y-7=0\end{array}\right.$$

Step-by-Step Solution

Answer

After graphing the equations and then algebraically confirming your results, you should find the correct intersection points (x, y).

Step 1: Graph the equations

Use relevant graphing software to graph both equations \(4x^2 - y^2 - 32x - 2y = -59\) and \(2x + y - 7 = 0\). Identify the point(s) of intersection.

Step 2: Substitute the x-values

Take the x-values from the intersection points and substitute them into both equations to determine the corresponding y-values.

Step 3: Confirm the solutions

Verify that these points of intersection are correct by ensuring the y-values from both equations are the same

Key Concepts

System of Equations

A system of equations is a collection of two or more equations that we deal with all together. In this case, we have one quadratic equation:

• \( 4x^2 - y^2 - 32x - 2y = -59 \)
• And one linear equation: \( 2x + y - 7 = 0 \)

We try to find a set of values for the variables, \(x\) and \(y\), that satisfy all equations in the system at the same time. This leads us to the point where the graphs of these equations intersect. Understanding this is crucial because such points often have important practical meanings, such as the solution to a physical problem or the regulator of a certain process.

Algebraic Verification

After finding the intersection points graphically, it's important to check their accuracy using algebra. Algebraic verification ensures that our graphical solution actually satisfies the original equations.Here's how we do it:

• From the graph, note the \(x\)-coordinates of the intersection points.
• Substitute these \(x\)-values back into both equations in the system.
• Solve for \(y\) in each case.
• Check if the \(y\)-values obtained are consistent for both equations.

If both equations are satisfied with the given \((x, y)\) pairs, you've successfully verified your solution algebraically. This step is essential because graphical methods can sometimes be inaccurate or imprecise.

Graphing Utility

A graphing utility is a tool, often software or a calculator, used to draw the graphs of functions and equations. In this problem, a graphing utility helps us identify where the system of equations intersects. Here's why it's beneficial:

• It offers a visual representation of mathematical concepts.
• Quickly finds intersection points, which can be complex to solve by hand.
• Handles complicated functions without extensive manual calculation.
• Contains features to zoom, trace, and adjust the view for maximum clarity.

Using technology in graphing not only aids in finding solutions but also enhances understanding of mathematical relationships by observing how equations behave graphically.

Quadratic Equation

In this problem, the quadratic equation is \( 4x^2 - y^2 - 32x - 2y = -59 \). Quadratic equations include terms where the variable is squared, resulting in a curved graph known as a parabola.Key features of quadratic equations:

• They can be expressed in the standard form \( ax^2 + bx + c = 0 \).
• The graph of a quadratic can open upwards or downwards.
• Solutions to quadratic equations can be real or complex numbers.
• The intersection points with other graphs depend on the degree and coefficients of each term.

Understanding quadratics is important because they appear frequently in various applications, from physics to economics, touching upon areas where variables influence each other nonlinearly.