Problem 37 - Algebra 1

Question

Write an equation of the line that is parallel to the given line and passes through the given point. $$y=-\frac{1}{3} x-1,(4,1)$$

Step-by-Step Solution

Answer

The equation of the line that is parallel to the given line and passing through the given point is \(y=-\frac{1}{3}x + \frac{7}{3}\).

Step 1: Identify the slope

The slope of the given line is given by the coefficient of 'x'. Thus, the slope of the line parallel to given line is \(m=-\frac{1}{3}\).

Step 2: Substitute in the point-slope form

Substitute the values of 'm' and the coordinates of the given point \((4,1)\) into the point-slope form equation. The equation thus becomes \(y - 1= -\frac{1}{3}(x - 4)\).

Step 3: Simplify the equation

Distribute the slope to the terms inside the parenthesis and simplify the equation to write it in slope-intercept form \(y=mx+c\). After simplification, equation of the line becomes \(y=-\frac{1}{3}x + \frac{7}{3}\).

Key Concepts

Point-Slope Form

The point-slope form is a versatile method of writing equations for lines, especially when a point on the line and the slope are known. The general form is expressed as
\( y - y_1 = m(x - x_1) \),
where \( (x_1, y_1) \) are the coordinates of the given point and \( m \) is the slope of the line. When using this form, it’s straightforward to plug in the slope and the coordinates to create the equation of a line.

For our exercise, we know the point \( (4,1) \) and that the slope of any line parallel to the given line is \( m=-\frac{1}{3} \). Substituting these values into the point-slope form, we get
\( y - 1 = -\frac{1}{3}(x - 4) \),
this is the equation of the line that has the same slope as the given line and passes through the point \( (4,1) \).

Slope-Intercept Form

The slope-intercept form has the structure
\( y = mx + c \),
where \( m \) represents the slope of the line and \( c \) is the y-intercept, the point where the line crosses the y-axis. This form is particularly useful because it immediately provides the two most defining characteristics of a straight line: its slope and where it intercepts the y-axis.

Once you’ve used the point-slope form to set up your equation, you can rearrange it to get the slope-intercept form, making it easier to graph and understand the behavior of the line. For the problem at hand, by distributing the slope and simplifying, we get the equation in slope-intercept form:
\( y = -\frac{1}{3}x + \frac{7}{3} \).
This tells us that our line goes down one unit for every three units it moves to the right and crosses the y-axis at \( \frac{7}{3} \).

Parallel Lines Properties

Properties of parallel lines are fundamental in understanding their behavior. Parallel lines never intersect, and they have the same slope. This is particularly important when writing equations for lines that are meant to be parallel to a given line. In the context of our example, since the given line has a slope of \( m=-\frac{1}{3} \), any line that is parallel to it must also have the same slope.

While the slopes are the same, the intercepts of parallel lines are generally different unless the lines are coincident (essentially, the same line). Hence, using the same slope but different points (or y-intercepts), one can generate infinite parallel lines, each with its own unique equation.

Identifying Slope

Identifying the slope is a key step in analyzing lines. The slope is a measure of the steepness or the inclination of a line and is calculated as the ratio of the rise to the run between any two points on a line — how much the line goes up or down for a horizontal movement. In an equation, the slope is usually represented by \( m \), and it is obtained from the coefficient of \( x \) in the slope-intercept form.

In our exercise, the slope is identified from the given equation of a line. Since the equation is \( y=-\frac{1}{3}x-1 \), the coefficient of \( x \) is \( -\frac{1}{3} \), which means that for each step to the right (positive \( x \) direction), the line goes down by \( \frac{1}{3} \) units in the \( y \) direction. This same value for the slope is used when writing the equation of a line parallel to it.