Problem 36 - A Graphical Approach to College Algebra

Question

A line having an equation of the form \(y=k x,\) where \(k\) is a real number, \(k \neq 0,\) will always pass through the origin. To graph such an equation by hand, we can determine a second point and then join the origin and that second point with a straight line. Use this method to graph each line. $$y=1.5 x$$

Step-by-Step Solution

Answer

The line \(y = 1.5x\) passes through the origin \((0,0)\) and point \((2,3)\).

Step 1: Understand the Given Equation

The given equation of the line is \(y = 1.5x\). This is in the form \(y = kx\), where \(k = 1.5\).

Step 2: Identify the Origin Point

The equation \(y = 1.5x\) implies that the line passes through the origin, which is the point \((0, 0)\).

Step 3: Determine a Second Point

To graph the line, choose a convenient value for \(x\). Let’s use \(x = 2\). Substitute \(x = 2\) into the equation: \(y = 1.5 \times 2 = 3\). So, the second point is \((2, 3)\).

Step 4: Plot the Points

On a graph paper or coordinate plane, plot the origin \((0, 0)\) and the second point \((2, 3)\).

Step 5: Draw the Line

Draw a straight line through the points \((0, 0)\) and \((2, 3)\). This line represents the equation \(y = 1.5x\).

Key Concepts

Slope-Intercept Form

To grasp linear equations efficiently, it's essential to understand the slope-intercept form. This particular format of a linear equation is given as \[ y = mx + b \] where:- \( m \) is the slope of the line.- \( b \) is the y-intercept, or where the line crosses the y-axis.
Notice that in our given equation, which is \( y = 1.5x \), we've got a slightly simpler version without the \( b \) term. This indicates that the line passes through the origin, which is why there’s no y-intercept in this case.
Recognizing this form allows us to easily pinpoint the slope, which is the factor that determines how steeply the line inclines or declines. In \( y = 1.5x \), the slope (\( 1.5 \)) tells us the rate of change for y as x increases by 1 unit.

Coordinate Plane

Plotting points accurately requires knowledge of the coordinate plane. Think of this plane as a big grid where you can plot exactly where you want your X and Y values to be. It has two axes:- The horizontal axis, known as the x-axis.- The vertical axis, called the y-axis.
These axes intersect at a central point called the origin, marked as \( (0, 0) \).
When graphing, consider the origin as your starting point. For the exercise, our equation \( y = 1.5x \) automatically passes through the origin. Both axes are instrumental in ensuring points are precise, as demonstrated by selecting \( x = 2 \), which corresponds to \( y = 3 \) and gives us the point \( (2, 3) \). By plotting this point beside the origin, we can draw an accurate representation of the line.

Straight Line Graphing

Creating a straight line graph can seem like magic, but it's just about connecting the dots! With our previous understanding of the slope and the coordinate plane, we can draw the line with confidence. Start by:

• Plotting the point where your line crosses (the origin, \( (0, 0) \) here).
• Finding and plotting a second point using the slope \( m \). In this instance, we calculated the point \( (2, 3) \).

Next, use a ruler to draw a straight line through these points. This line visually represents all possible \( x \) and \( y \) values that satisfy the equation \( y = 1.5x \). The line is infinite, but only the visible segment offers a glimpse of the entire solution set. This straightforward exercise exemplifies the predictability and precision of linear equations.