Problem 29 - Algebra 1: Concepts and Skills

Question

Write in standard form an equation of the line that passes through the given point and has the given slope. \((7,3), m=-\frac{1}{3}\)

Step-by-Step Solution

Answer

The standard form of the line that passes through the point (7,3) and has a slope of \(-\frac{1}{3}\) is \(x + 3y = 16\).

Step 1: Insert the given values into the point-slope form

The given point is \((7,3)\) and the slope is \(-\frac{1}{3}\). Therefore, \(x1 = 7\), \(y1 = 3\), and \(m = -\frac{1}{3}\). Substituting these values into the point-slope form equation, we get: \(y - 3 = -\frac{1}{3} (x - 7)\)

Step 2: Simplify the equation

Multiply through by 3 to eliminate the fraction: \(3(y - 3) = -1(x - 7)\). This simplifies to \(3y - 9 = -x + 7\).

Step 3: Write in standard form

To convert the equation into standard form, put the terms involving \(x\) and \(y\) on the same side of the equation. So, the equation becomes \(x + 3y = 16\).

Key Concepts

Point-Slope Form

The point-slope form of a line equation is a powerful tool. It helps us find the equation of a line if we know its slope and a point on the line. The general formula is given by:

• \( y - y_1 = m(x - x_1) \)

In this formula:

• \( (x_1, y_1) \) represents the coordinates of the known point.
• \( m \) is the slope of the line.

When we substitute the values from our exercise, where the point is \((7,3)\) and the slope \(m = -\frac{1}{3}\), we get:
\( y - 3 = -\frac{1}{3}(x - 7) \).
Starting with the point-slope form allows us to easily manipulate the equation into other forms, like the standard form.

Slope

The slope of a line is a measure of its steepness or angle. We often denote it using the letter \(m\). The slope can tell us how the line rises or falls as it moves from left to right. Mathematically, the slope is calculated as:

• \( m = \frac{\Delta y}{\Delta x} \)

Here, \( \Delta y \) represents the change in the vertical direction, and \( \Delta x \) is the change in the horizontal direction.
In our example, the slope is \(-\frac{1}{3}\), indicating a negative slope:

• A negative slope means the line descends as it moves from left to right.
• A slope of \(-\frac{1}{3}\) implies that for every 3 units you move horizontally to the right, the line falls by 1 unit vertically.

Understanding slope is crucial for constructing and interpreting line equations.

Line Equation

A line equation provides a mathematical representation of a line on a graph. Simply put, it shows us how to calculate \(y\) for any given value of \(x\). Line equations can be expressed in different forms including:

• Point-slope form: Used when we know a point and the slope of the line.
• Slope-intercept form: \( y = mx + b \), where \(b\) is the y-intercept.
• Standard form: \( Ax + By = C \), where \(A\), \(B\), and \(C\) are integers.

For our exercise, we've been tasked to convert the line equation into standard form. From point-slope form, we performed algebraic operations to rearrange the terms.
The result was \( x + 3y = 16 \). This is the line's equation in standard form, helping us to understand its entire structure at a glance.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that connects algebra and geometry. It involves graphing and understanding geometric figures using coordinates on the plane.

• Points are represented by their coordinates: \((x, y)\).
• Lines are represented by equations, such as point-slope or standard form.

In our exercise, we used a given point and slope to form an equation that represented a specific line on the coordinate plane.
The main advantage of coordinate geometry is that it allows us to analyze geometric figures through algebraic equations. This makes it easier to visualize and solve complex geometric problems. By converting line equations from one form to another, we're using the principles of coordinate geometry to express relationships in a more convenient format for further analysis or application.