Problem 59 - 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. $$\left\\{\begin{array}{l}x-y+3=0 \\ x^{2}-4 x+7=y\end{array}\right.$$

Step-by-Step Solution

Answer

The points of intersection can be found visually through graphing, and then confirmed algebraically through substitution.

Step 1: Graph the Equations

Use a graphing utility to plot the two equations \(x - y + 3 = 0\) and \(x^{2} - 4x + 7 = y\). This will provide a visual representation of the equations, allowing for the identification of the intersection points.

Step 2: Identify Intersection Point

Examine the graph to locate the point(s) where the two curves intersect. These represent the solution(s) to the system of equations.

Step 3: Algebraic Confirmation

Substitute one equation into the other to confirm the intersection point identified during the graphing process. In this case, \(x - y + 3 = 0\) can be rearranged to \(y = x + 3\). Substituting this equation into the second equation \(x^{2} - 4x + 7 = y\) and simplifying allows one to verify the intersection coordinates algebraically.

Key Concepts

Graphing utility

A graphing utility is a tool often used to visualize mathematical equations. In this case, it helps you plot the graphs of the given equations effortlessly. Graphing utilities are available as:

• Online graphing calculators
• Software applications like GeoGebra and Desmos
• Physical devices such as TI-84 calculators

A graphing utility provides a visual representation of equations, making it easier to understand where and how different equations intersect.
For our exercise, you would plot the equations \( x - y + 3 = 0 \) and \( x^{2} - 4x + 7 = y \). This means you would draw the two lines or curves on the same axes. The plots will display intersection points, which indicate the values of \( x \) and \( y \) that satisfy both equations. Visualizing these intersections clarifies the solutions immediately, as opposed to solely relying on algebraic methods.

System of equations

A system of equations consists of two or more equations with common variables. The goal is to find the set of values that satisfy all equations in the system at the same time. In our example, the system is:

• \( x - y + 3 = 0 \)
• \( x^{2} - 4x + 7 = y \)

Each equation in the system provides a condition which the solution must fulfill.
The intersection of graphs visually represents the point(s) that satisfy both equations. These points form the solution to the system. With two equations in one system, both lines or curves need to be considered. You must identify their common point(s).
If plotted correctly, the graphs will visually highlight these intersecting points direcly.

Algebraic confirmation

After identifying the intersection points using a graphing utility, algebraic confirmation solidifies your answer. This step is crucial to ensure accuracy and reliability.
Start by rearranging the first equation \( x - y + 3 = 0 \) to express \( y \) in terms of \( x \): \[ y = x + 3 \]
Substitute \( y = x + 3 \) into the second equation \( x^{2} - 4x + 7 = y \). Doing so replaces \( y \) with the expression derived from the first equation: \[ x^{2} - 4x + 7 = x + 3 \]
Simplify the equation: \[ x^{2} - 4x + 7 - x - 3 = 0 \] \[ x^{2} - 5x + 4 = 0 \]
Solve this quadratic equation, often through factoring, completing the square, or using the quadratic formula:

• Factor as \((x - 1)(x - 4) = 0\), hence \( x = 1 \) or \( x = 4 \)
• For \( x = 1, \, y = 1 + 3 = 4 \)
• For \( x = 4, \, y = 4 + 3 = 7 \)

The solutions \((1, 4)\) and \((4, 7)\) match the intersection points found using the graphing utility, confirming your answer is correct both visually and algebraically.