Problem 8 - A Graphical Approach to College Algebra

Question

Write the slope-intercept form of the line that passes through the given point with slope \(m .\) Do not use a calculator. Through \(\left(5,-\frac{1}{3}\right), m=3\)

Step-by-Step Solution

Answer

The equation of the line is \(y = 3x - \frac{46}{3}\).

Step 1: Recall the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept of the line.

Step 2: Substitute the Given Slope

We know the slope \(m\) of the line is 3. Substitute 3 for \(m\) in the slope-intercept form equation. Now the equation looks like \(y = 3x + b\).

Step 3: Use the Given Point to Find the Y-intercept

The line passes through the point \( (5, -\frac{1}{3}) \). Substitute \(x = 5\) and \(y = -\frac{1}{3}\) into the equation \(y = 3x + b\) to find \(b\). This gives:\(-\frac{1}{3} = 3(5) + b\).

Step 4: Solve for the Y-intercept

Solve the equation \(-\frac{1}{3} = 15 + b\) for \(b\). First, subtract 15 from both sides:\(-\frac{1}{3} - 15 = b\),Convert 15 to a fraction with denominator 3: \(= \frac{-1}{3} - \frac{45}{3} = \frac{-1 - 45}{3} = \frac{-46}{3}\).Thus, \(b = -\frac{46}{3}\).

Step 5: Write the Final Equation

Now that you have both \(m = 3\) and \(b = -\frac{46}{3}\), substitute back into the slope-intercept form. The equation of the line is \(y = 3x - \frac{46}{3}\).

Key Concepts

Equation of a Line

The equation of a line is a mathematical expression that describes a straight line, which is the shortest distance between any two points in plane geometry. They convey essential information, such as the slope and intercepts, which help us understand the position and steepness of the line. Most commonly, lines are expressed in the slope-intercept form, point-slope form, or standard form.

• Slope-Intercept Form: This is one of the most widely used forms, given by the formula \( y = mx + b \) where \( m \) is the slope of the line, and \( b \) is the y-intercept.
• Point-Slope Form: This form expresses a line using a known point on the line and its slope: \( y - y_1 = m(x - x_1) \).
• Standard Form: Given as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are real numbers.

Each of these forms has its use cases, and converting between them involves straightforward algebraic manipulations. The slope-intercept form is particularly useful because it provides a quick way to find the direction and starting point of a line on a graph.

Linear Equations

Linear equations are a fundamental type of equation in algebra that describes a straight line. These equations are linear because they only involve linear terms, which are variables raised to the first power.
Linear equations can be identified easily due to their form: they typically appear as expressions that equate a linear polynomial on one side to a constant or another linear polynomial. The simplest form can be written as \( ax + by = c \) where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables.
A key feature of linear equations is that they graph to straight lines. The slope of the line is determined by the coefficient of \( x \), and the y-intercept is the constant term in the equation when rewritten in slope-intercept form \( y = mx + b \). Solving linear equations involves finding the values of the variables for which the equation holds true, and often entails:

• Isolating the variable on one side of the equation.
• Performing arithmetic operations to simplify.

They are not only used theoretically in mathematics but also have practical applications in fields like economics and engineering, where relationships between variables can often be described linearly.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, deals with points and lines on a coordinate plane. It combines algebra and geometry to give a robust toolset for solving geometrical problems by representing geometric figures using coordinates.
One of the powerful aspects of coordinate geometry is its ability to represent every point uniquely through coordinates \((x, y)\) on the Cartesian plane. This representation allows for precise calculations and the ability to visualize geometrical concepts through plotting.

Understanding Points and Lines

• Points: Represented as \((x, y)\), where \(x\) is the horizontal coordinate, and \(y\) is the vertical coordinate.
• Lines: Described by linear equations that can be transformed between different forms, such as slope-intercept, to draw on the coordinate plane.
• Slope: The measure of the steepness of a line. Calculated as \( m = \frac{y_2 - y_1}{x_2 - x_1} \) using two points \((x_1, y_1)\) and \((x_2, y_2)\).

Coordinate geometry not only helps in solving theoretical problems but is also widely useful in real-world applications involving graphs, maps, and various computational geometry tasks.