Problem 27 - Algebra 1

Question

\(.\) Logical REASONING A line with a positive slope passes through the origin, making a \(60^{\circ}\) angle with the positive \(x\) -axis. Write an equation of the line.

Step-by-Step Solution

Answer

The equation of the line is \(y = \sqrt{3}x\).

Step 1: Find the slope

The slope \(m\) of the line can be found using the fact that the tangent of the angle the line makes with the x-axis is equal to the slope. The angle in this case is \(60^{\circ}\), hence \(m = \tan(60^{\circ})\).

Step 2: Calculate the slope

Using the tangent value for \(60^{\circ}\), which is \(\sqrt{3}\), find the slope: \(m = \tan(60^{\circ}) = \sqrt{3}\).

Step 3: Write the equation of the line

Substitute the calculated slope \(m\) into the slope-intercept form of the line equation \(y = mx + b\). Since the line passes through the origin, \(b\) is 0. Hence the equation of the line is \(y = \sqrt{3}x\).

Key Concepts

Slope-intercept form

The slope-intercept form is a common way to express the equation of a straight line. This form is written as \( y = mx + b \).
- **\( y \)** represents the dependent variable or the value along the vertical axis.- **\( x \)** is the independent variable, plotted along the horizontal axis.- **\( m \)** is the slope of the line, indicating how steep the line is. A positive \( m \) means the line slopes upward.- **\( b \)** is the y-intercept, the point where the line crosses the y-axis.When writing an equation in slope-intercept form, we need to identify the slope \( m \) and the y-intercept \( b \). In our original exercise, it was given that the line passes through the origin. Passing through the origin means that the y-intercept \( b \) is 0. Therefore, once we determine the slope, we can construct the entire equation using the form \( y = mx \).

Tangent of an angle

Tangent is a trigonometric function that relates an angle of a right triangle to the lengths of its two legs. For a given angle \( \theta \), the tangent is defined as the ratio of the opposite side length to the adjacent side length.
In the context of lines, the tangent of the angle a line makes with the x-axis gives us the slope of that line, which is very helpful when the angle is known, as this method allows us to directly calculate the slope. For the angle of \( 60^{\circ} \), the tangent is \( \sqrt{3} \), which simply means that a line making a \( 60^{\circ} \) angle with the x-axis ascends at a steepness quotient of about 1.73 (since \( \sqrt{3} \approx 1.73 \)). Therefore, for our problem, the slope \( m \) of the line is this tangent value: \( m = \sqrt{3} \).

Positive slope

A positive slope represents a line that rises as it moves from left to right on a graph. This means as we increase the x-value, the y-value will also increase. Such a line moves in an upward direction, showing a positive relationship between the variables.
This concept is vital in numerous fields requiring an understanding of directional change and trend analysis.

• In mathematics and physics, a positive slope signifies increasing values or growth.
• In economics, it could represent rising profits or other benefits with increased input or time.

In our exercise, because the given angle with the x-axis is \( 60^{\circ} \), and it is within the first quadrant, it indicates the line ascends, confirming its positive slope. This positive slope is found using \( m = \tan(60^{\circ}) = \sqrt{3} \), ready to be used in our line equation.