Problem 42 - College Algebra and Calculus: An Applied Approach

Question

Solve the system graphically. $$\left\\{\begin{aligned}-x+2 y &=1 \\ x-y &=2 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution to the system is the point where the two lines intersect, which is (2, 0).

Step 1: Rearrange Equations

First, both equations shall be rearranged into the slope-intercept form, which is \(y = mx + c\), where m is the slope and c is the y-intercept. Rearranging, the two equations become: \(y = 0.5x + 0.5\) and \(y=x-2\)

Step 2: Graph the Equations

Next step involves plotting these two equations on our graph to visually represent them. Start by marking the y-intercepts and then using the slope to determine additional points. After plotting a sufficient number of points, draw a line to represent the equation.

Step 3: Identify the Intersection

The third step is to identify where the two lines intersect, if they do at all. This point is the solution to the system because it is the value for x and y that makes both equations true simultaneously. In this case, they intersect at the point (2, 0).

Key Concepts

System of Equations

A system of equations is a collection of two or more equations that share common variables. When solving a system of equations, we aim to find a set of values for these variables that satisfy all the equations simultaneously. There are multiple methods to solve these systems, including substitution, elimination, and graphical methods. As in our exercise, solving a system graphically involves plotting each equation on a graph and finding the point where the lines intersect. This point represents a common solution to all equations in the system.

• A system is considered consistent if there is at least one solution.
• An inconsistent system has no solution.
• If there are infinitely many solutions, the system is dependent.

Understanding how to set and interpret systems of equations is fundamental in algebra, as it lays the groundwork for more complex mathematical concepts.

Slope-Intercept Form

The slope-intercept form is a common way of expressing linear equations. It is written as \( y = mx + c \), where \( m \) represents the slope of the line, and \( c \) is the y-intercept. Using this form makes it easy to quickly graph a line, as the y-intercept tells us where the line crosses the y-axis, while the slope indicates how steep the line is.To convert an equation into slope-intercept form, you might need to rearrange the terms so that \( y \) is isolated on one side of the equation. For example, given the equation \( -x + 2y = 1 \), you can solve for \( y \) to obtain \( y = 0.5x + 0.5 \).

• The slope \( m \) is the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
• The y-intercept \( c \) is where the line crosses the y-axis when \( x = 0 \).

The slope-intercept form is particularly useful for determining how a line behaves and for finding the intersection point with other lines.

Intersection Point

The intersection point of two lines on a graph is a crucial concept, particularly when solving systems of equations graphically. This point is where the two lines meet or cross, and it represents a solution because it satisfies both equations simultaneously.To find the intersection point, first, ensure each line is plotted on the same set of axes after they are rearranged into the slope-intercept form. In our example, the equations \( y = 0.5x + 0.5 \) and \( y = x - 2 \) intersect at the point \( (2, 0) \). This point is interpreted as the solution to the system.

• The intersection point shows the values of \( x \) and \( y \) that meet the criteria of both equations.
• Determining this point graphically means checking where the lines on the graph cross each other.
• If the lines are parallel and do not intersect, the system has no solution.
• If the lines overlap completely, there are infinitely many solutions.

Recognizing and understanding the significance of the intersection point is key in solving linear systems using the graphical method.