Problem 3 - Algebra A Combined Function

Question

Write an equation of each line with the given slope and containing the given point. Write the equation in the slope-intercept form \(y=m x+b .\) See Example \(1 .\) Slope \(-2 ;\) through (1,-3)

Step-by-Step Solution

Answer

The equation of the line is \( y = -2x - 1 \).

Step 1: Identify the Given Values

We are given the slope \( m = -2 \) and a point \( (x_1, y_1) = (1, -3) \). First, we'll use these values to find the equation of the line in slope-intercept form \( y = mx + b \).

Step 2: Use the Point-Slope Formula

The point-slope form of a line is \( y - y_1 = m(x - x_1) \). Substitute \( m = -2 \), \( x_1 = 1 \), and \( y_1 = -3 \) into the equation.\[ y + 3 = -2(x - 1) \]

Step 3: Simplify the Equation

Expand the equation from Step 2:\[ y + 3 = -2x + 2 \]Subtract 3 from both sides to solve for \( y \):\[ y = -2x + 2 - 3 \]\[ y = -2x - 1 \]

Step 4: Write the Equation in Slope-Intercept Form

The equation in slope-intercept form is \( y = -2x - 1 \). This matches the format \( y = mx + b \) with \( m = -2 \) and \( b = -1 \).

Key Concepts

Linear Equations

A linear equation is a mathematical expression that forms a straight line when graphed. The general form of a linear equation in two variables, usually written as \( x \) and \( y \), is \( y = mx + b \). This equation represents a line through a coordinate plane and includes two key components: the slope \( m \) and the y-intercept \( b \).
Understanding linear equations is crucial because they describe a steady rate of change. In many real-world situations, they model consistent relationships between two variables.

• **Slope**: The incline or steepness of the line.
• **Y-intercept**: The point where the line crosses the y-axis.

Recognizing the format and components of a linear equation helps in constructing, understanding, and graphing lines effectively.

Slope

The slope of a line is a measure of its steepness or incline. It's often denoted by the letter \( m \). The mathematical definition of slope is the change in the y-value divided by the change in the x-value between two points on the line.
This is also known as "rise over run" and is calculated as:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]For example, if we take two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope tells us how much y changes when x increases by 1 unit.

• A positive slope means the line inclines upwards.
• A negative slope means the line declines downwards.
• A zero slope indicates a completely horizontal line.

Understanding the slope is essential when graphing lines and analyzing the relationship between x and y in a linear context.

Point-Slope Form

The point-slope form is a useful way to express a line equation when you know the slope and a single point on the line. It is expressed as:\[ y - y_1 = m(x - x_1) \]Here, \((x_1, y_1)\) is the point on the line, and \(m\) is the slope. This form allows you to write an equation quickly without needing the y-intercept.
It's particularly handy when you need to derive the equation of a line when given a slope and a point that's not necessarily on the y-axis.

• Write down the slope you are given.
• Plug in the coordinates of the point provided.
• Simplify to convert to slope-intercept form if needed.

The point-slope form bridges the gap between raw data and the familiar slope-intercept form, making it invaluable in solving and understanding line equations.

Graphing Lines

Graphing a line from an equation is a way to visually interpret its behavior on the coordinate plane. When dealing with equations in slope-intercept form \( y = mx + b \), the process becomes straightforward.
Start by identifying the slope \( m \) and y-intercept \( b \).

• **Y-intercept (b)**: Locate this point on the y-axis; it's where your line will cross.
• **Using the Slope (m)**: From the y-intercept, apply the slope by "rising" and "running" to plot another point. If \( m = \frac{3}{2} \), go up 3 units and right 2 units.

Draw the line through these points. Remember:

• A positive slope will angle upwards as you move from left to right.
• A negative slope will angle downwards.

This visual representation helps in grasping the relationship between variables and seeing potential intersection points with other lines.