Problem 84 - College Algebra Essentials

Question

Exercises \(83-85\) will help you prepare for the material covered in the first section of the next chapter. Graph \(x+2 y=2\) and \(x-2 y=6\) in the same rectangular coordinate system. At what point do the graphs intersect?

Step-by-Step Solution

Answer

The graphs of \(x + 2y = 2\) and \(x - 2y = 6\) intersect at the point (2,-1).

Step 1: Express the equations in y = mx + c form

The given equations are \(x + 2y = 2\) and \(x - 2y = 6\). We need to write these equations in the 'y = mx + c' form. The transformed equations become:\(y = -0.5x + 1\) and \(y = 0.5x - 3\).

Step 2: Graph the equations

Using any graphing tool or paper and pencil, we graph these two lines on the same coordinate system. You can start by plotting the y-intercepts and then using the slope to find another point on each line.

Step 3: Find the Intersection

The point of intersection is the point where the two lines cross. Solving the system of linear equations \(y = -0.5x + 1\) and \(y = 0.5x - 3\), we find this point to be (2, -1).

Step 4: Summary of Results

From the graph plot or by applying system of linear equations, it can be found that the two lines intersect at the point (2, -1). Thus, the point of intersection of the given two lines is (2,-1).

Key Concepts

Coordinate System

The coordinate system is a fundamental tool for graphing and understanding linear equations. It consists of two perpendicular lines, the X-axis (horizontal) and the Y-axis (vertical), which intersect at a point called the origin. Each point on this plane can be described by an ordered pair \( (x, y) \) representing its coordinates, with \( x \) denoting the position along the horizontal axis and \( y \) indicating the position along the vertical axis.

For instance, when graphing the linear equations \( x+2y=2 \) and \( x-2y=6 \) from our exercise, we use this coordinate system to plot points and draw lines. By mapping out several points for each equation and then connecting these points, we form a visual representation of the equations as straight lines on the graph.

Slope-Intercept Form

Slope-intercept form is the equation of a line written as \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept. The slope represents the steepness and the direction of the line, while the y-intercept represents the point where the line crosses the Y-axis.

In the solution provided, we have converted both equations to this form: \( y = -0.5x + 1 \) and \( y = 0.5x - 3 \) which makes it easier to graph and understand the characteristics of the lines. The slope \( -0.5 \) indicates that the first line is falling (descending from left to right), and \( 0.5 \) suggests the second line is rising (ascending from left to right). The y-intercepts are \( 1 \) and \( -3 \) for the respective lines, providing a starting point for plotting each line on a graph.

Systems of Linear Equations

Systems of linear equations involve finding the solution for two or more lines when considered together. Solving a system of linear equations means finding the point(s) at which the lines intersect, that is, the set of coordinates that satisfy all the equations simultaneously.

From the given problem, we have a system composed of the equations \( y = -0.5x + 1 \) and \( y = 0.5x - 3 \). Solving this system entails determining the exact point where these two lines meet. Various methods can be used, such as graphing, substitution, elimination, or using matrices. In the context of our exercise, graphing provided the visual solution, and algebraic methods could confirm the point of intersection.

Point of Intersection

The point of intersection of two lines is the point where the lines cross over each other on the graph. It signifies the solution to the system of linear equations represented by the lines. This point has the unique property that its coordinates satisfy the equations of both lines.

In the exercise, by graphing or solving the equations \( y = -0.5x + 1 \) and \( y = 0.5x - 3 \) we determined that the point of intersection is \( (2, -1) \). This means that when \( x \) is 2, the \( y \) value for both lines is \( -1 \)—they meet at the same place. Identifying the point of intersection is crucial in many fields, including algebra, calculus, and even real-life applications such as navigation and economics.