Problem 21 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((1,7) ;\) slope \(\frac{2}{3}\)

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{2}{3}x + \frac{19}{3} \).

Step 1: Understand the Point-Slope Form

The point-slope form of the equation of a line is given by \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope of the line. In this problem, the point is \((1, 7)\) and the slope is \(\frac{2}{3}\).

Step 2: Substitute Point and Slope into Point-Slope Form

Substitute \(x_1 = 1\), \(y_1 = 7\), and \(m = \frac{2}{3}\) into the point-slope formula. This gives:\[ y - 7 = \frac{2}{3}(x - 1) \]

Step 3: Simplify the Equation

Distribute \(\frac{2}{3}\) on the right-hand side of the equation:\[ y - 7 = \frac{2}{3}x - \frac{2}{3} \]Then, solve for \(y\) by adding 7 to both sides to put the equation in slope-intercept form \(y = mx + b\):\[ y = \frac{2}{3}x - \frac{2}{3} + 7 \]

Step 4: Simplify the Constant Term

Convert 7 into a fraction with denominator 3 to combine it with \(-\frac{2}{3}\):\[ 7 = \frac{21}{3} \]So the equation becomes:\[ y = \frac{2}{3}x + \left( -\frac{2}{3} + \frac{21}{3} \right) \]

Step 5: Finalize the Equation

Add the constants:\[ -\frac{2}{3} + \frac{21}{3} = \frac{19}{3} \]Thus, the final equation of the line is:\[ y = \frac{2}{3}x + \frac{19}{3} \]

Key Concepts

Point-Slope Form

The point-slope form is a particularly useful way to write the equation of a line when you have one point and the slope. This formula shows the line's steepness (slope) and how it behaves around a specific point. The general equation is given by:

• \( y - y_1 = m(x - x_1) \)

Here, \(x_1, y_1\) is a point on the line, and \(m\) is the slope. This format highlights the change in \(y\) with respect to \(x\) around the point \(x_1, y_1\).
For instance, if we know a line goes through the point \( (1, 7) \) and has a slope of \frac{2}{3}\, we can substitute these values into the formula:

• \( y - 7 = \frac{2}{3}(x - 1) \)

The point-slope form makes it easy to write the line's equation quickly without calculating further points or values. Understanding this form helps a lot when dealing with line equations in coordinate geometry.

Slope-Intercept Form

The slope-intercept form is another common way of writing line equations, providing clear insight into both the slope and the y-intercept. This form is expressed as:

• \( y = mx + b \)

In this equation, \(m\) represents the slope, while \(b\) stands for the y-intercept, which is the point where the line crosses the y-axis. For clarity, if we have transformed a point-slope equation like \( y - 7 = \frac{2}{3}x - \frac{2}{3} \) into slope-intercept form, we solve for \( y \), resulting in:

• \( y = \frac{2}{3}x + \frac{19}{3} \)

This equation informs us that the line has a slope of \frac{2}{3}\ and it crosses the y-axis at \frac{19}{3}\. The slope-intercept form is incredibly helpful for graphing lines and understanding their behavior overall.

Slope

The slope of a line is a measure of its steepness, often described as "rise over run." It is denoted by the letter \( m \), and is computed by the change in y-coordinates divided by the change in x-coordinates between two points on the line. Mathematically, it is expressed as:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

A positive slope, like \frac{2}{3}\, indicates that the line rises as it moves from left to right. A negative slope means the line falls. A slope of zero corresponds to a horizontal line, and an undefined slope is characteristic of a vertical line. Understanding slope is crucial as it's a fundamental attribute of lines in coordinate geometry, detailing how lines interact in a plane.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebraic techniques to study geometry. It allows us to find different properties of geometric figures, like distance, midpoint, and the equation of lines. By assigning coordinates to points on a plane, problems in geometry become manageable with algebra.
In this context, the line's equation through \( (1, 7) \) with a slope of \frac{2}{3}\ is a prime example of how coordinates and algebra combine to describe geometric figures with precision.
Coordinate geometry is powerful because it provides a method to solve many geometry problems accurately, which would otherwise be complex using purely geometric methods. It offers a way to visualize equations and solve intersection, parallelism, and perpendicularity problems effectively.