Problem 10 - Algebra A Combined Function

Question

Solve each system of linear equations by graphing. See Examples 3 through \(6 .\) \(\left\\{\begin{array}{l}x+y=3 \\ x-y=5\end{array}\right.\)

Step-by-Step Solution

Answer

The solution is where the graphs intersect: \( (4, -1) \).

Step 1: Identify the Equations

We have two linear equations that we need to solve by graphing:1. Equation 1: \( x + y = 3 \)2. Equation 2: \( x - y = 5 \).

Step 2: Find the domain and intercepts

Determine the domain, x-intercepts, and y-intercepts.

Step 3: Analyze asymptotes and end behavior

Find vertical, horizontal, and oblique asymptotes.

Step 4: Find critical points and intervals

Compute the derivative, find critical points, and determine increase/decrease intervals.

Step 5: Summarize the graph

Combine all information to describe the graph.

Step 6: Conclude with the answer

The solution is where the graphs intersect: \( (4, -1) \).

Key Concepts

Graphing Method

The graphing method is a way of solving systems of equations by plotting the equations on a graph. It provides a visual representation of the solution. Each equation in the system is graphed on the same set of axes, and the point where the graphs intersect represents the solution of the system.

To graph an equation, you generally need to rewrite it in slope-intercept form, which is \( y = mx + b \). Here, \( m \) is the slope, and \( b \) is the y-intercept. For instance, let's transform the equations from our exercise:

• Equation 1: \( x + y = 3 \) can be rewritten as \( y = -x + 3 \).
• Equation 2: \( x - y = 5 \) becomes \( y = x - 5 \).

The intersection point of these lines on a graph represents the x and y values that are solutions to both equations at the same time. In this particular exercise, when you graph these two lines, they intersect at the point \( (4, -1) \), which is the solution to the system. The method is straightforward and particularly useful when dealing with two-variable systems.

Linear Equations

A linear equation is an equation of a straight line, which can be written in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. In simpler terms, it's any equation that graphs as a straight line. These equations have clear solutions that can often be easily determined through graphing.

In the provided exercise, the equations \( x + y = 3 \) and \( x - y = 5 \) are linear because they describe straight lines when graphed. To understand linear equations thoroughly, it’s crucial to consider their general properties:

• Slope: This is the rate at which the line rises or falls. It is denoted by \( m \) in the slope-intercept form.
• Intercept: The y-intercept, \( b \), is the point where the line crosses the y-axis.
• Solution: A solution of a linear equation is any point \( (x, y) \) that satisfies the equation.

Understanding these concepts is vital when graphing and finding intersections since these characteristics will guide how and where you plot the lines.

Systems of Equations

Systems of equations involve solving two or more equations that share common variables. The main objective is to find values for the variables that satisfy all equations in the system simultaneously. In general terms, solving a system of equations means finding the intersection and harmonious solution of simultaneously applied conditions.

In our exercise, we deal with the system of equations:

• \( x + y = 3 \)
• \( x - y = 5 \)

The graphing method allows us to find the point of intersection of these two lines, which is the common solution or, more specifically, the pair \( (x, y) \) that satisfies both equations. You place each equation on a graph and look for where they meet. Here, that happens at the point \( (4, -1) \).

Besides graphing, other methods include substitution and elimination, which rely more heavily on algebraic manipulation rather than visual representation. Understanding the purpose and characteristics of each method, as well as the nature of the equations involved, helps in efficiently solving these systems.