Problem 63 - Introductory Algebra for College Students

Question

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-x+2$$

Step-by-Step Solution

Answer

The five solutions for the equation \(y = -x + 2\) are (-2,4),( -1, 3), (0, 2), (1, 1), (2, 0). When graphed, these solutions form a straight line with a negative slope.

Step 1: Constructing the Table

To find the points on the graph, substitute values of \(x\) in the equation to find corresponding \(y\) values. Start the tabulation by choosing a negative, a zero and positive values for \(x\). Try \(x\) = -2, -1, 0, 1, 2. Plug these into the equation and get the following: when \(x = -2\), \(y = -(-2) + 2 = 4\); when \(x = -1\), \(y = -(-1) + 2 = 3\); when \(x = 0\), \(y = -(0) + 2 = 2\); when \(x = 1\), \(y = -(1) + 2 = 1\); when \(x = 2\), \(y = -(2) + 2 = 0\). This gives the pairs (-2,4),( -1, 3), (0, 2), (1, 1), (2, 0).

Step 2: Plotting the Graph

Plot these points on a graph: (-2,4), (-1, 3), (0, 2), (1, 1), (2, 0). Each pair represents Cartesian coordinates, with the first number indicating the x-coordinate and the second number the y-coordinate. To plot a point, start at the origin (0,0) and move right or left for \(x\) units and up or down for \(y\) units. After plotting the points, draw a line through them to complete the graph. The line will have a negative slope and a positive intercept on the y-axis, which concurs with the equation given.

Key Concepts

Cartesian Coordinates

In the realm of graphing and plotting data, the Cartesian coordinate system plays a crucial role. This system uses two perpendicular lines: the horizontal line known as the x-axis and the vertical line known as the y-axis.

Together, they form a grid that allows you to pinpoint locations using ordered pairs, or coordinates.

Each pair is represented as \((x,y)\). For instance, the coordinate \((-2, 4)\) signifies a point two units to the left of the origin and four units up.

• The origin is the point where both axes intersect, marked as \((0,0)\).
• The x-coordinate tells you how far left or right to move from the origin.
• The y-coordinate shows how far up or down to move from there.

Understanding how to identify and plot points using Cartesian coordinates is fundamental in graphing linear equations.

Once you grasp this grid system, plotting points becomes a straightforward task of navigating through the x and y axes.

Table of Values

Constructing a table of values is a foundational step towards graphing linear equations. This method helps you generate specific points that the line will pass through. To start, select a series of x-values.

Commonly, a mix of negative, zero, and positive numbers is used. Then plug these values into your equation to calculate the corresponding y-values.

• In the example given with \(y=-x+2\):
• Choosing \(x = -2, -1, 0, 1, 2\) leads to:\((x,y)\) pairs of \((-2,4), (-1,3), (0,2), (1,1), (2,0)\).

This process will give you ordered pairs that define points on your graph.

Think of the table of values as a translator turning your equation into clear points on a graph. Make sure to calculate at least a few points, as this helps ensure your line is accurately represented across the graph's plane.

Slope-Intercept Form

The slope-intercept form is a popular way to express linear equations, like the one in our example. It is written as \(y = mx + b\), where \(m\) represents the slope, and \(b\) denotes the y-intercept.

This form is advantageous because it instantly reveals the slope and where the line crosses the y-axis.

• In \(y = -x + 2\), the slope \(m = -1\), indicating a downward slant of the line.
• The y-intercept \(b = 2\) tells us the line hits the y-axis at \((0,2)\).

Understanding slope-intercept form simplifies graphing because you get a quick overview of the line's behavior. The slope gives direction and steepness, while the intercept sets the starting point on the y-axis. This format is a versatile tool for graphing and analyzing linear relationships, ensuring clear and efficient conversion from equation to graph.