Problem 37 - Calculus with Concepts in Calculus

Question

Find an equation of the line that is parallel to the given line \(l\) and passes through the given point \(P\). \(l: y=3 x-1 ; P=(2,-1)\)

Step-by-Step Solution

Answer

The equation of the parallel line is \( y = 3x - 7 \).

Step 1: Understanding Parallel Lines

Parallel lines have the same slope. The given line is \( y = 3x - 1 \), which means its slope \( m \) is 3. So, any line parallel to this will also have the slope \( m' = 3 \).

Step 2: Apply Point-Slope Form

We use the point-slope form of a line equation: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is the given point. Using point \( P = (2, -1) \) and slope \( m = 3 \), we apply it to find: \( y + 1 = 3(x - 2) \).

Step 3: Simplify to Slope-Intercept Form

Distribute the 3 on the right side: \( y + 1 = 3x - 6 \). Then, isolate \( y \) by subtracting 1 from both sides: \( y = 3x - 7 \). This is the equation of the line parallel to line \( l \) and passing through point \( P \).

Key Concepts

Slope of a Line

The slope of a line is a measure that describes how steep a line is. You might think of it as the 'tilt' of the line. In mathematical terms, slope is often represented by the letter \( m \) and is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The formula for calculating slope \( m \) when given two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

• When the slope is positive, the line rises as it moves from left to right.
• If the slope is negative, the line falls as it moves from left to right.
• A zero slope means the line is horizontal.
• An undefined slope, often encountered when dividing by zero, indicates a vertical line.

In the exercise, the line \( l: y = 3x - 1 \) has a slope of 3. This means that for every 1 unit of increase in \( x \), \( y \) increases by 3 units. Any other line parallel to this will share this same slope.

Point-Slope Form

The point-slope form is a convenient way to write the equation of a line when you know the slope and one point on the line. It's especially handy when you are dealing with parallel lines, or when given a point to create an equation. The point-slope equation is written as: \[ y - y_1 = m(x - x_1) \]
Here, \( m \) represents the slope, and \((x_1, y_1)\) is a point on the line. For our exercise, we're using the point \( P = (2, -1) \) and the slope \( m = 3 \).
Let's see how the formula works:

• Plug the slope \( m = 3 \) into the equation.
• Use the coordinates \( (x_1, y_1) = (2, -1) \).

Now substitute these values: \[ y + 1 = 3(x - 2) \]. This equation represents a line parallel to the original line \( l \), passing through the point \( P \).
The point-slope formula is versatile and can be rearranged into other forms for graphing or analysis.

Slope-Intercept Form

The slope-intercept form is another popular way to express the equation of a line, known for its straightforwardness. It simplifies understanding where a line crosses the y-axis and its slope. This form is: \[ y = mx + b \]
Where \( m \) is the slope and \( b \) is the y-intercept (the point where the line crosses the y-axis).
To convert from the point-slope form to the slope-intercept form:

• Start with the point-slope equation.
• Simplify and isolate \( y \).

In the exercise solution, the point-slope equation \( y + 1 = 3(x - 2) \) was expanded to \( y = 3x - 7 \), by distributing \( 3 \) and adjusting the terms. This results in a straight line equation that tells you instantly that the slope \( m \) is 3, and the y-intercept \( b \) is -7. This form is helpful for graphing lines or quickly understanding their behavior.