Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find the slope and y-intercept (if possible) of the equation of the line algebraically, and (b) sketch the line by hand. Use a graphing utility to verify your answers to parts (a) and (b). $$5 x-y+3=0$$

Step-by-Step Solution

Answer

The slope of the line is 5 and the y-intercept is 3.

Step 1: Convert to Slope-Intercept Form

It is required to convert the given equation \(5x - y + 3 = 0\) into the slope-intercept form \(y = mx + c\). This can be done by firstly isolating y. The equation then becomes: \(5x + 3 = y\)

Step 2: Identify the Slope and Y-Intercept

In the equation, compare it with \(y = mx + c\). Therefore, \(m\), the slope of the line, is equal to 5 and \(c\), the y-intercept, is equal to 3.

Step 3: Sketch the Line

Plots the y-intercept on a grid. From there, use the slope to determine a second point and draw the line that passes through these two points.

Step 4: Verify with Graphing Utility

Using a graphing tool, input the original equation and verify that the graph obtained is consistent with the manual sketch,and it has the same slope and y-intercept as determined in the previous steps.

Key Concepts

Slope-Intercept Form

Linear equations can be represented in various forms, but one of the most intuitive is the slope-intercept form. This is expressed as \( y = mx + c \), where \( y \) is the dependent variable, \( m \) represents the slope, and \( c \) is the y-intercept.
To convert an equation into slope-intercept form, you need to solve for \( y \). This structure makes it easy to quickly identify both the slope and the y-intercept just by looking at the equation.
For example, convert the equation \( 5x - y + 3 = 0 \) to \( y = 5x + 3 \). Here, the slope \( m \) is 5, and the y-intercept \( c \) is 3.

Graphing Linear Equations

Graphing linear equations starts by plotting the y-intercept on the graph. From the slope-intercept form, once we've established the y-intercept, it's your starting point on the y-axis.
Next, use the slope to find another point. Slope is essentially the "rise over run." In our equation \( y = 5x + 3 \), the slope \( m \) is 5, or \( \frac{5}{1} \), meaning from the y-intercept point, you move up 5 units and 1 unit to the right.

• Plot the y-intercept (0,3) on the graph.
• From there, count up 5 units and 1 unit to the right to find the next point.

Draw a straight line through these points, and you have successfully graphed your equation. Always ensure your points are correctly aligned, and use a ruler to maintain precision.

Identifying Slope and Y-Intercept

Identifying the slope and y-intercept of a linear equation is straightforward once it's in the slope-intercept form \( y = mx + c \). The slope \( m \) describes how steep the line is and in which direction it leans. If \( m \) is positive, the line tilts upwards; if negative, downwards.
The y-intercept \( c \) locates where the line crosses the y-axis. It's an important feature that gives a quick visual cue on graphs.

• For \( 5x - y + 3 = 0 \), transform to \( y = 5x + 3 \).
• The slope is 5, showing the line rises steeply.
• The y-intercept is 3, indicating where it crosses the y-axis.

This identification allows us to quickly and accurately graph the line or evaluate its relation to other lines.