Problem 38 - Precalculus with Limits

Question

Solving a System of Equations Graphically In Exercises \(33-44,\) solve the system graphically. $$\left\\{\begin{aligned}-x+y &=3 \\ x^{2}-6 x-27+y^{2} &=0 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations will be the points where the line and the circle intersect. Some geometrical skills will be needed to find the exact intersection points on the graph paper or one can use graphing tools to get the precise points.

Step 1: Graphing the Linear Equation

The first equation is a linear equation which can be rewritten in slope-intercept form (y=mx+b) as: \( y = x + 3 \). Now plot the line.

Step 2: Graphing the Quadratic Equation

The second equation is a quadratic equation in terms of x and y. It can be rewritten as \( x^{2} - 6x + y^{2} +27 = 0 \), which is in the form of equation of a circle \( (x-h)^{2} + (y-k)^{2} = r^{2} \) where center is \((h,k)\) and radius is \(r\). Now plot the circle on the same graph as the line.

Step 3: Finding Intersection Points

The solutions to this system of equations are the points where the line and the circle intersect. Find these points on the graph which is plotted.

Key Concepts

Linear Equations

A linear equation is a type of algebraic equation whose graph is a straight line. It can be written in several forms, but a common one is the slope-intercept form: \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) signifies the y-intercept, which is the point where the line crosses the y-axis.
Linear equations are characterized by a few key points:

• They have a constant rate of change, indicated by the slope \( m \).
• They produce a graph shaped as a straight line.
• The degree of the variables is always one.

In our exercise, we rewrote the first equation \( -x + y = 3 \) into the slope-intercept form as \( y = x + 3 \). This simplification allows us to easily identify the slope and intercept, helping us graph the line correctly.

Quadratic Equations

Quadratic equations involve an equation of second degree, usually in the form \( ax^2 + bx + c = 0 \). However, they can also appear in different formats, including those involving both \( x \) and \( y \), forming figures like parabolas or even circles, as in our problem.
Some defining characteristics of quadratic equations include:

• The highest degree of the variable is two.
• They can be represented as parabolas if dependent on a single variable or circles if written as \((x-h)^2 + (y-k)^2 = r^2\), as in the exercise.

For this exercise, the quadratic equation \( x^2 - 6x - 27 + y^2 = 0 \) can be rearranged to form a circle. Solving graphically involves plotting this on the coordinate plane alongside the linear equation.

System of Equations

A system of equations consists of two or more equations with the same set of variables. The objective is to find the solution that satisfies all the equations simultaneously. Systems can be solved using various methods, but graphical solutions are particularly intuitive, especially when involving distinct shapes like lines and circles.
In general, a system can have:

• No solution, if the graphs do not intersect.
• One solution, if they touch at a single point.
• Infinitely many solutions, if they are identical lines or fully overlapping shapes.

In our problem, the system consists of one linear and one quadratic equation, with solutions represented by the intersection points of these graphs.

Slope-Intercept Form

The slope-intercept form of a linear equation, \( y = mx + b \), is an efficient way to quickly determine the graph's key characteristics. This form directly provides:

• The slope \( m \) of the line, showing how steeply the line inclines or declines.
• The y-intercept \( b \), indicating where the line crosses the y-axis.

In plotting or graphing, this format is advantageous because it makes identifying the starting point on the y-axis and the inclination of the line straightforward. For our given linear equation \( -x + y = 3 \), converting to \( y = x + 3 \) allowed us to easily draw it on a graph for visual analysis.

Intersection Points

Intersection points are the coordinates where two or more graphs meet on a coordinate plane. They represent solutions to systems of equations when graphed. In our task, identifying intersection points between the plotted line and circle is the goal.
The methodology for finding intersection points involves:

• Graphing each equation accurately on the same coordinate axes.
• Observing where the graphs cross each other.
• Determining the exact coordinates of these crossing points by reading the graph or solving the system algebraically for precision.

In our exercise, these points indicate the solutions to the system \( \{ y = x + 3, \; x^2 - 6x + y^2 - 27 = 0 \} \), where each point satisfies both the linear and quadratic equations.