Problem 19 - Elementary Algebra

Question

For the following problems, graph the equations. $$ y-5 x+4=0 $$

Step-by-Step Solution

Answer

Answer: The slope of the equation is $$5$$, and the y-intercept is $$-4$$. To graph it, first plot the y-intercept point $$(0, -4)$$ on the y-axis. From this point, use the slope to move up $$5$$ units and right $$1$$ unit to plot the next point. Continue this process to plot additional points, and then connect these points with a straight line. This line represents the graph of the given equation.

Step 1: Rewrite the equation in slope-intercept form

To change the given equation $$y - 5x + 4 = 0$$ into slope-intercept form, we need to isolate $$y$$ on one side of the equation. We can do this by simply adding $$5x$$ and subtracting $$4$$ from both sides: $$ y = 5x - 4 $$

Step 2: Determine the slope and y-intercept

Now that the equation is in slope-intercept form, we can identify the slope $$m$$ and y-intercept $$b$$: $$ m = 5\\ b = -4 $$ The slope is $$5$$ and the y-intercept is $$-4$$.

Step 3: Plot the graph

To plot the graph, follow these steps: 1. Begin by plotting the y-intercept $$(-4)$$ on the y-axis, which is the point $$(0, -4)$$. 2. Use the slope $$5$$ (which, as a fraction, is $$5/1$$) to count the rise over the run from the y-intercept point. Since it's a positive slope, this means you'll go up $$5$$ units and right $$1$$ unit. 3. Continue this process to plot additional points on the graph, and plot at least two more points for accuracy. 4. Finally, connect these points to draw a straight line – the graph of the given equation: $$y = 5x - 4$$. Now you have successfully graphed the equation $$y - 5x + 4 = 0$$.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is essential for graphing it accurately. The slope-intercept form is written as
\( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) indicates the y-intercept. The slope, \( m \), describes how steep the line is and the direction it tilts – upward or downward, while the y-intercept, \( b \), tells us where the line crosses the y-axis.

By rewriting the equation from the exercise
\( y - 5x + 4 = 0 \) into the slope-intercept form, we get
\( y = 5x - 4 \). Having the equation in this format simplifies the entire process of graphing as it provides clear indicators: the slope of 5 and the y-intercept of -4. Knowing these two parameters, one can immediately begin to plot the line on a graph.

Slope and Y-Intercept

As soon as the equation is in slope-intercept form, identifying the slope (\( m \)) and the y-intercept (\( b \)) becomes straightforward. The slope of \( 5 \), or \( 5/1 \) when expressed as a fraction, dictates the sharpness of the angle of the line and its direction. A positive slope, like in our example, implies that the line rises from left to right.

The y-intercept at \( b = -4 \) gives us a starting point on the graph. It is where the line will intersect the y-axis. In coordinate terms, this point is \( (0, -4) \). With both the slope and y-intercept determined from \( y = 5x - 4 \), plotting the rest of the graph becomes a matter of using these two pieces of information effectively.

Plotting Points

Plotting points is the concrete action you take to mark specific locations on a graph corresponding to solutions of the equation. Starting with the y-intercept, plot the point \( (0, -4) \) on the y-axis. This is your anchor. From this point, the slope tells you how to move to plot additional points. With our slope of 5, you move up 5 units (since the slope is positive) for every 1 unit you move to the right, ending at another valid point for the line.

To ensure accuracy when sketching the line, it is wise to plot at least two more points using the same rise-over-run technique. Each point solidifies the accuracy of your graph. After marking these points, use a ruler to draw a straight line through them, which represents the graph of your linear equation. This visualization helps solidify the relationship between the algebraic formula and the geometric representation on the graph.