Problem 26 - Calculus: One and Several Variables

Question

Find an equation for the line that passes through the point (2,-3) and is parallel to the line \(y=1\)

Step-by-Step Solution

Answer

The equation of the line that passes through the point (2,-3) and is parallel to the line \(y=1\) is \(y = -3\).

Step 1: Identify the Slope of the Given Line

The given line \(y=1\) is a horizontal line. The slope of any horizontal line is 0. Therefore, the slope (m) of the line parallel to \(y=1\) is also 0.

Step 2: Apply the slope-intercept form of equation

Parallel lines have equal slopes so our line will share slope 0 with \(y=1\). Now, we will use the slope-intercept equation of a line which is \(y = mx + b\), where m is the slope and b is the y-intercept. Substituting the slope m=0 and the point (2, -3) into this formula we get \(-3 = 0*2 + b\). Solving for b we get \(b = -3\).

Step 3: Write out the equation of the line

Now, to write out the equation for our line, we substitute m = 0 (slope) and b = -3 (y-intercept) into the equation \(y = mx + b\). So, the equation for the line that passes through (2, -3) and is parallel to \(y=1\) is \(y = -3\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a line equation is a fundamental concept in algebra. This form is expressed as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept, the point where the line crosses the y-axis.

The beauty of the slope-intercept form lies in its directness; it tells us the steepness of the line (slope) and the line's crossing point on the y-axis (y-intercept) at a glance. To find the equation of a line given a slope and a point, as in our exercise, we simply plug in the given values into the formula and solve for \( b \

Applying the Slope-Intercept Form

When we know a point on the line, \(x_1, y_1\)\), and the slope, we calculate the y-intercept by rearranging our formula to solve for \( b \), which leads to: \( b = y_1 - mx_1 \). In our solved example, we substituted \( m = 0 \) and \( (2, -3) \) into the formula, found that our \( b \) was \( -3 \), and therefore the line's equation became \( y = -3 \).

Parallel Lines

Parallel lines are lines in a plane that never intersect, meaning they have the same slope. When we're tasked with finding a line parallel to another, we first determine the slope of the reference line and use that same slope for the new line.

This property simplifies the process of finding equations of parallel lines, as we already have one of the key components – the slope. In our exercise, the slope of the already given horizontal line \( y=1 \) is zero. Therefore, the line passing through the point \( (2, -3) \) and parallel to \( y=1 \) also has a slope of zero.

Identifying Parallel Lines

Whenever two linear equations have the same slope, they are parallel. However, if their y-intercepts differ, they'll never meet - as was the case with our original line \( y=1 \) and the sought line \( y=-3 \).

Horizontal Line Slope

The slope of a horizontal line is always zero because horizontal lines have no vertical change as you move along the line. The 'rise' over 'run' calculation for slope, or \( \frac{\text{change in y}}{\text{change in x}} \), results in \( 0 \) since the change in y (rise) is zero for horizontal lines.

Remember, a horizontal line is always in the form \( y = k \), where \( k \) is the constant y-value for all points on the line. For instance, the line given in the exercise, \( y=1 \), indicates a horizontal line where every point on this line has a y-coordinate of 1.

Characteristics of Horizontal Lines

Not only do horizontal lines serve as a great reference for understanding the concept of zero slope, but they also come into play during graphical analysis, such as understanding that if a line is parallel to the x-axis, it's a horizontal line. The line we found in our exercise, \( y = -3 \), is another example of a horizontal line, bearing the hallmark slope of zero.