Problem 45 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of line that passes through the given points. \(m=-2, b=3\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = -2x + 3\)

Step 1: Identify the slope and y-intercept

In the problem, we are given that the slope \(m\) is -2 and the y-intercept \(b\) is 3.

Step 2: Substitute the slope and y-intercept into the equation

The general form of the equation of a line is \(y = mx + b\). Substitute the given slope and y-intercept into this equation to get \(y = -2x + 3\).

Key Concepts

Equation of a Line

In mathematics, an equation of a line describes a straight path on a two-dimensional graph. The equation acts as a set of instructions for plotting points along that line. Usually, when calculating the equation of a line, you use additional information such as the slope and y-intercept, which help in drawing the line accurately.

Typically, the equation of a line is expressed in different forms, but the slope-intercept form is the most common. In this form, the equation looks like this:

• \( y = mx + b \)

- The letter \( 'y' \) stands for the dependent variable, while \( 'x' \) is the independent variable.
- \( 'm' \) represents the slope of the line, indicating how steep the line is.
- \( 'b' \) is the y-intercept, showing where the line crosses the y-axis.Understanding the parts of this equation helps you easily graph the line it represents. It also makes calculations more straightforward, particularly when predicting other points on the same line.

Slope

The slope is a measure of how steep a line is on a graph. It tells you how much the line rises or falls for each unit of movement to the right. In the slope-intercept form, it is represented by the letter \( 'm' \).
The slope is calculated using the formula:

• \( m = \frac{\text{change in } y}{\text{change in } x} \)

This formula explains the rate of change between any two points on a line. Let's look at the slope value:\(-2\). This tells us that for every step you move right along the x-axis, the line moves down by 2 units on the y-axis.

A positive slope means the line goes up, and a negative slope means the line goes down. The larger the absolute value of the slope, the steeper the line.

Y-Intercept

The y-intercept is a critical part of the equation of a line. It is the point where the line crosses the y-axis. In the slope-intercept form, this is denoted by the letter \( 'b' \). Ordinarily, it appears at the very end of the equation \( y = mx + b \).
For instance, a y-intercept of \(3\) signifies that the line meets the y-axis at the point \( (0, 3) \). This tells you where the line starts on the vertical axis.

Knowing the y-intercept is helpful, especially when you need to sketch a graph quickly. It gives you a starting point from which you can plot other points using the slope. Thus, in constructing or interpreting equations of lines, the y-intercept provides a clear reference, ensuring that you're plotting the path correctly.