Problem 51 - Brief Applied Calculus

Question

Show that \(y-y_{1}=m\left(x-x_{1}\right)\) simplifies to \(y=m x+b\) if the point \(\left(x_{1}, y_{1}\right)\) is the \(y\) -intercept \((0, b)\).

Step-by-Step Solution

Answer

It simplifies to \(y = mx + b\) when substituting \(x_1 = 0\) and \(y_1 = b\).

Step 1: Understanding the Given Equation

The given equation is the point-slope form of a line: \(y-y_1=m(x-x_1)\). This equation represents a line with a slope \(m\) passing through the point \((x_1, y_1)\).

Step 2: Substitute the Y-Intercept Point

If the point \((x_1, y_1)\) is the y-intercept \((0, b)\), we substitute \(x_1 = 0\) and \(y_1 = b\) into the equation: \(y-b=m(x-0)\).

Step 3: Simplify the Equation

The substituted equation \(y-b=m(x-0)\) simplifies to \(y-b=mx\). Since \(x-0\) is \(x\), the equation becomes \(y-b=mx\).

Step 4: Solve for y

Add \(b\) to both sides of the equation to solve for \(y\): \(y = mx + b\). This is the slope-intercept form of the equation of a line.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line's equation is one of the most commonly used representations, making it incredibly important in algebra. It's represented as \( y = mx + b \). Here, \( m \) stands for the slope of the line, which indicates how steep the line is. Meanwhile, \( b \) represents the y-intercept, the point where the line crosses the y-axis. This form is popular because it's straightforward and provides direct information about the line's key characteristics.

• Slope \( m \): Determines the angle and direction of the line. A positive slope means the line rises, while a negative slope indicates it falls.
• Y-intercept \( b \): The starting point of the line on the y-axis when \( x = 0 \). It's where the line 'intercepts' the y-axis.

Remember, the slope-intercept form is specifically suited for making quick graphs of lines as you automatically get the y-intercept and can easily apply the slope to find additional points.

Y-Intercept

In a linear equation, the y-intercept is a crucial component because it reveals the initial value of the function when \( x = 0 \). This means the y-intercept is the height at which the line crosses the y-axis.When the equation is in slope-intercept form \( y = mx + b \), \( b \) is the y-intercept. It provides a starting point for graphing the line.Key points to consider when identifying and interpreting the y-intercept:

• A y-intercept at \( (0, b) \) means when you start plotting your line on the y-axis, this is the point you'll begin with.
• It's often used in real-world problems, such as budgeting or physics, where it represents a fixed cost or initial speed.

By knowing the y-intercept, you can immediately place the first point on the y-axis, making it much simpler to complete the rest of the graph using the slope.

Equation of a Line

An equation of a line is a mathematical way to describe a straight line on a graph. In algebra, we often use various forms to represent this equation, making it easier to work with in different situations.Common forms of a linear equation include:

• Point-Slope Form \( y-y_1=m(x-x_1) \): This form is useful when you have a known point and the slope and want to write the equation of the line passing through that point.
• Slope-Intercept Form \( y = mx + b \): Perfect when you know the slope and y-intercept, making it ideal for graphing.
• Standard Form \( Ax + By = C \): Often used for linear algebra and systems of equations.

By understanding how to work with each form, you can solve problems, graph lines, and understand how equations describe real-world scenarios. Transitioning between these forms, like simplifying the point-slope form to the slope-intercept form, further enhances your problem-solving skill set in algebra.