Problem 36 - Algebra 1

Question

Write an equation of the line that is parallel to the given line and passes through the given point. $$y=\frac{2}{3} x-2,(2,1)$$

Step-by-Step Solution

Answer

The equation of the line parallel to \(y = \frac{2}{3}x -2\) and passing through the point (2,1) is \(y = \frac{2}{3}x - \frac{1}{3}\).

Step 1: Identify the Slope of the Given Line

From the given equation \(y = \frac{2}{3}x -2\), the slope \(m\) is \frac{2}{3}

Step 2: Use the Slope and Given Point to Find the New Y-intercept

We substitute the slope and the given point (2,1) into the equation \(y = mx + c\) to find \(c\), the y-intercept. Substituting, we get \(1 = \frac{2}{3}(2) + c . Solving for c gives us c = -\frac{1}{3}\)

Step 3: Write the Equation of the New Line

Finally, substituting the slope and y-intercept into the equation \(y = mx + c\) gives us the equation of the new line: \(y = \frac{2}{3}x - \frac{1}{3}\)

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial when dealing with linear equations. The slope is a numerical measure of how steep a line is. In mathematical terms, it represents the rate of change of the y-coordinate with respect to the x-coordinate of any two distinct points on the line.

The slope, often denoted as 'm', is calculated by dividing the difference in y-coordinates by the difference in x-coordinates of these points. For the line equation in the form \(y = mx + b\), 'm' is the slope.

Calculating Slope from a Linear Equation

Look directly at the coefficient of the x-term in the equation. For instance, in the equation \(y = \frac{2}{3}x - 2\), the slope is \(\frac{2}{3}\). This means that for every 3 units the line moves horizontally (along the x-axis), it moves up or down by 2 units vertically (along the y-axis).

Slope and Parallel Lines

Lines that are parallel always have the same slope. Hence, to write the equation of a line parallel to another, you simply use the same slope as the given line.

By using the slope, we can understand and predict the behavior of the line, and it provides the foundation for many concepts in algebra and calculus.

Point-Slope Form

When you have a point and the slope of a line, you can use the point-slope form to write its equation. The point-slope form of the equation of a line is expressed as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \( (x_1, y_1) \) is the point the line passes through.

Applying Point-Slope Form

To apply this form, you'll first need to know the slope of your line and the coordinates of the given point. The exercise provided shows the process of using a known point (2,1) and the slope \(\frac{2}{3}\) to find the equation of a parallel line.

It's important to note that while the point-slope form is extremely useful for writing equations quickly, it's often useful to simplify the equation to slope-intercept form \(y = mx + b\) to make the graphing and interpretation of the line easier.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis on a graph. It's represented by the letter 'b' in the slope-intercept form of a linear equation, \(y = mx + b\). The y-intercept is a key aspect in graphically representing equations as it provides a starting point for the line on the graph.

Finding the Y-intercept

In some cases, like when using point-slope form, you'll have to rearrange the equation to solve for 'b', which is the y-intercept. This can be accomplished by substituting the slope and the coordinates of a point that the line passes through into the slope-intercept form and solving for b.

In the given exercise, after determining the slope is \(\frac{2}{3}\), substituting the point (2,1) yields the y-intercept \(\frac{-1}{3}\). Consequently, the line that is parallel to \(y = \frac{2}{3}x - 2\) and passes through the point (2,1) is written as \(y = \frac{2}{3}x - \frac{1}{3}\). Recognizing the y-intercept in a linear equation allows for immediate identification of one point on the line, making the process of graphing simpler.