Problem 17 - Elementary Algebra

Question

For the following problems, graph the equations. $$ 2 x-y+4=0 $$

Step-by-Step Solution

Answer

Answer: The graph of the linear equation \(2x - y + 4 = 0\) is a straight line with a slope of 2 and intercept of -4. Sample points on the line include (-2, 0), (-1, 2), (0, -4), (1, -6), and (2, -8).

Step 1: Rearrange the equation to solve for y

The given equation is: $$ 2x - y + 4 = 0 $$ To solve for y, we will isolate y on one side of the equation: $$ -y = -2x + 4 $$ Multiply both sides by -1 to make y positive: $$ y = 2x - 4 $$ Now, the equation is in slope-intercept form (y = mx + b), which makes it easier to graph.

Step 2: Create a table of values

To create the table, we pick several values for x and plug them into the rearranged equation to find the corresponding y values. x | y -2 | 0 -1 | 2 0 | -4 1 | -6 2 | -8

Step 3: Plot the points and draw the line

Now, plot each (x, y) point from the table on the Cartesian plane: (-2, 0) (-1, 2) (0, -4) (1, -6) (2, -8) After plotting the points, draw a straight line that connects each point. This line represents the graph of the equation \(y = 2x - 4\).

Key Concepts

Slope-Intercept Form

When it comes to graphing linear equations, understanding the slope-intercept form is crucial. It is an equation of the form \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept, or the point where the line crosses the y-axis. In the given exercise, the slope-intercept form of the equation \( 2x - y + 4 = 0 \) has been derived as \( y = 2x - 4 \).

Here, the slope \( m \) is 2, indicating that for every one unit increase in x, y increases by two units. The y-intercept \( b \) is -4, telling us that the line crosses the y-axis at \( (0, -4) \). By identifying these two components, students can quickly plot the y-intercept and use the slope to determine the direction and steepness of the line. This form simplifies graphing by offering a clear starting point and a consistent way to locate other points on the line.

Table of Values

A table of values is a practical tool for graphing linear equations. It involves choosing input values (usually x-values), calculating the corresponding output (y-values), and listing them in a two-column table. The exercise involved creating such a table to find points through which the linear equation passes.

For example, using the equation \( y = 2x - 4 \), various x-values were selected, and their corresponding y-values were computed. These points, such as (0, -4), (1, -6), and so on, each represent a coordinate on the Cartesian plane that lies on the graph of the equation. Plotting each of these points and then connecting them can reveal the overall graph. It's essential to pick x-values that are distributed across the range of interest and to ensure calculation accuracy while transforming these x-values into y-values.

Cartesian Plane

The Cartesian plane is a foundational concept in graphing linear equations. It is a two-dimensional plane divided by a horizontal x-axis and a vertical y-axis, which intersect at the origin (0,0). Each point on the plane is defined by an ordered pair (x, y) that represents its coordinates.

To graph an equation on the Cartesian plane, one uses a set of points that satisfy the equation, such as those found using the table of values. In the exercise, the points (-2, 0), (-1, 2), and so on, were plotted. Students should carefully place each point according to its coordinates and then use a ruler to draw a line through these points, making sure the line extends in both directions, demonstrating the infinite set of solutions that lie on this line. The Cartesian plane enables visualization of the relationship between the x and y variables as defined by the equation.