Problem 1 - 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 \(3 ;\) through (1,2)

Step-by-Step Solution

Answer

The equation of the line is \(y = 3x - 1\).

Step 1: Understanding the Problem

You are given a line that passes through the point (1,2) and has a slope of 3. You need to write the equation of this line in the slope-intercept form, which is given as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Substitute Point into Equation

Using the slope-intercept form, \(y = mx + b\), substitute the x-value, y-value, and slope from the given point and slope. With (1,2) and \(m = 3\), the equation becomes \(2 = 3(1) + b\).

Step 3: Solve for b

We need to solve for \(b\), the y-intercept. Start by calculating \(3(1)\), which is 3, and then subtract 3 from both sides of the equation: \(2 = 3 + b\) becomes \(b = 2 - 3\). Thus, \(b = -1\).

Step 4: Write the Equation

Substitute the slope \(m = 3\) and the calculated y-intercept \(b = -1\) back into the slope-intercept form \(y = mx + b\). The equation of the line is \(y = 3x - 1\).

Key Concepts

Equation of a Line

An equation of a line is a mathematical representation of a straight line. It's usually expressed in various forms. One popular form is the slope-intercept form, written as \(y = mx + b\). Here:

• \(y\) represents the dependent variable
• \(x\) represents the independent variable
• \(m\) is the slope of the line
• \(b\) is the y-intercept.

To form the equation of a line, you need two components: the slope and a point on the line. Using these, you can substitute into the slope-intercept form to find the y-intercept. This process helps in deriving the complete equation of the line. Understanding how each part of the equation works helps in interpreting the graph of the line and making predictions based on it.

Solving for Y-Intercept

The y-intercept (\(b\)) is a crucial part of the slope-intercept form of a line equation. It's the point where the line crosses the y-axis. To find the y-intercept, you must know the slope and at least one point on the line. Let's break down the steps needed to determine the y-intercept:

• First, insert the slope \(m\) and the coordinates of the point (x, y) you know into the equation \(y = mx + b\).
• Next, solve for \(b\), the y-intercept, by isolating it on one side of the equation.

For example, if you have a slope of 3 and a point (1, 2), substitute these values into the equation, resulting in \(2 = 3 \cdot 1 + b\). Simply solve for \(b\) by performing the arithmetic to find \(b = -1\). Once you have \(b\), you can complete the line's equation, aiding in graph drawing or data analysis.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using a coordinate plane to study geometric figures. It bridges algebra and geometry by describing shapes with algebraic equations. The main components are:

• The x-axis and y-axis, which form the coordinate plane.
• Points, represented as coordinates like (x, y).
• Lines, described by equations such as the slope-intercept form \(y = mx + b\).

This mathematical area is essential for analyzing line equations. Given a point and a slope, you can use algebra to determine the equation of the line. The line's equation reveals vital information, like how its slope indicates the line's steepness. Understanding coordinate geometry enables you to visualize lines and curves and solve problems associated with distances, midpoints, and angles.