Problem 24 - Algebra 1

Question

Use the graphing method to solve the linear system and tell how many solutions the system has. $$ \begin{aligned}&x+y=8\\\&x+y=-1\end{aligned} $$

Step-by-Step Solution

Answer

The system has 0 solutions as the given lines are parallel and do not intersect.

Step 1: Expression of the Equations in Slope-Intercept Form

First, need to write the given equations in the slope intercept form, \(y = mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. The provided equations are already in this format.\n\nEquation 1: \(y = -x + 8\)\n\nEquation 2: \(y = -x - 1\)

Step 2: Graphing of the Equations

Each of these equations represent a line on a graph. Plot these to get a vivid image of what's happening. The graph will have two straight lines. The line \(y = -x + 8\) intercepts the y-axis at the point (0,8), and the line \(y = -x - 1\) intercepts the y-axis at the point (0,-1). It's clear that both lines are parallel and never meet.

Step 3: Determining Number of Solutions

For a system of linear equations to have a solution, the lines represented by the equations must intersect at a point. In this case, the two lines are parallel and do not intersect, which means this system of equations does not have a solution. Therefore, the system has 0 solutions.

Key Concepts

Graphing Method

The graphing method is a visual technique used to solve systems of linear equations. By representing each equation as a line on a graph, we can determine if the lines intersect at a point. Here’s how it usually works:

• Graph each equation: Convert the equations into lines on the same graph. This involves plotting points that satisfy each equation and then connecting them.
• Analyze the graph: Check if the lines intersect. The point of intersection represents the solution to the system of equations.

This method provides a visual approach, making it easier to understand the behavior of the equations. However, it's crucial to note that graphical solutions are approximate unless the point of intersection clearly lies on grid lines. In the example problem, when using the graphing method, the lines do not intersect, indicating no solution.

Solutions of Linear Systems

The solutions of linear systems refer to the points that satisfy all equations involved. For systems with two equations, there are three possible outcomes when plotted on a graph:

• One Solution: The lines intersect at one point. This point represents the values of variables that satisfy both equations simultaneously.
• No Solution: The lines are parallel and never meet. They have the same slope but different y-intercepts, as seen in our example.
• Infinite Solutions: The lines lie on top of each other, meaning they are the same line with all points in common.

In our specific linear system, the lines are parallel, indicating no solutions. This outcome is a critical aspect when solving linear systems, as it tells us whether there is compatibility or conflict between the equations.

Slope-Intercept Form

The slope-intercept form is a way of expressing linear equations as: \[ y = mx + b \] Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, where the line crosses the y-axis.
This form is useful for graphing because:

• Easy to plot: You can quickly sketch the line by starting at the y-intercept and following the slope to find further points.
• Identifies parallel lines: If two lines have the same \( m \) (slope), they are parallel and will never meet.

In our exercise, both equations were already in slope-intercept form, making plotting simpler. The lines had the same slope, indicating parallelism and, consequently, no solutions to the system.