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Chapter 2: Problem 5

Graph each equation. Check your work. $$ 5 x-2 y=-4 $$

Short Answer

Expert verified
The graph of the equation 5x - 2y = -4 is a straight line with a slope of 5/2 and a y-intercept at (0,2). It can be plotted by starting at the y-intercept and using the slope to find another point such as (2,7), then drawing a line through these points.

Step by step solution

01

Convert the equation to slope-intercept form

To graph the equation, it will be helpful to rewrite it in slope-intercept form, which is given by y = mx + b, where m is the slope and b is the y-intercept. Starting with the given equation, 5x - 2y = -4, we need to solve for y. Add 2y to both sides to isolate the y terms on one side: 5x = 2y - 4. Next, add 4 to both sides to isolate the constant term: 5x + 4 = 2y. Finally, divide every term by 2 to solve for y: y = (5/2)x + 2.
02

Identify the slope and y-intercept

Now that the equation is in slope-intercept form, y = (5/2)x + 2, we can identify the slope (m) and the y-intercept (b). The slope, m, is the coefficient of x, which is 5/2. The y-intercept, b, is the constant term, which is 2.
03

Plot the y-intercept on the graph

Begin plotting the graph by marking the y-intercept. The y-intercept is the point where the line crosses the y-axis. Since the y-intercept is 2, plot the point (0,2) on the graph.
04

Use the slope to plot another point

With the slope of 5/2, starting from the y-intercept (0,2), move up 5 units and to the right 2 units to determine a second point on the line. Plot this point on the graph, which will be (2,7).
05

Draw the line

Finally, draw a straight line through the two points (0,2) and (2,7). This line represents the graph of the equation 5x - 2y = -4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
Understanding the slope-intercept form is fundamental for graphing linear equations. It is written as y = mx + b, where m represents the slope of the line, and b indicates the y-intercept. The beauty of this form is its directness; it immediately tells us how steep the line is and where it crosses the y-axis.

In the given equation, 5x - 2y = -4, we rearrange terms to put it into slope-intercept form. By isolating y on one side, the equation becomes y = (5/2)x + 2. This format makes plotting the line on a graph straightforward, as it simplifies identifying the slope and y-intercept directly from the equation.
Identifying Slope and Y-Intercept
To properly graph a linear equation, pinpointing the slope and y-intercept is crucial. The slope, denoted as m in the equation y = mx + b, indicates the line's tilt. It is the change in y (rise) over the change in x (run). A positive slope means the line ascends rightward, while a negative slope inclines it leftward.

The y-intercept, represented by b, is the point where the line crosses the y-axis, given by the coordinates (0, b). In our equation, y = (5/2)x + 2, the slope is 5/2, signifying a rise of 5 units for every run of 2 units to the right. The y-intercept is 2, determining the starting point on the graph at (0,2). Identifying these two components is the first step before laying the line out on the graph.
Plotting Points on a Graph
With the slope and y-intercept in hand, plotting a line on a coordinate graph is a clear-cut process. Start by drawing the y-intercept. For our example where the y-intercept is 2, you would place a dot at (0,2) because the graph crosses the y-axis at this point.

Following this, use the slope to find another point. With a slope of 5/2, begin at the y-intercept and move vertically up 5 units and horizontally to the right 2 units to reach the new point, (2,7). Mark this second dot on the graph. Now, simply connect these dots with a straight line, extending the line evenly in both directions. This drawn line is the visual representation of the linear equation 5x - 2y = -4. By plotting a graph in this way, complex algebraic expressions are translated into clear visual data, making it easier for you to analyze and understand.

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