Problem 14 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((-3,2)\) with slope \(-1\)

Step-by-Step Solution

Answer

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

Step 1: Understand 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. In this problem, the slope \( m \) is given as \(-1\) and the line passes through the point \((-3,2)\).

Step 2: Substitute slope into the equation

Substitute the given slope \(-1\) into the slope-intercept form equation: \[ y = -1x + b \] or \[ y = -x + b \]

Step 3: Substitute the point to find the y-intercept

Substitute the point \((-3,2)\) into the equation to solve for \( b \). When \( x = -3 \), \( y = 2 \). So, \[ 2 = -(-3) + b \] Simplify to find \( b \): \[ 2 = 3 + b \] \[ b = 2 - 3 \] \[ b = -1 \]

Step 4: Write the final equation

Substitute \( b = -1 \) back into the equation: \[ y = -x - 1 \]

Key Concepts

Slope-Intercept Form

In algebra, one of the most common ways to express the equation of a line is the slope-intercept form. This form is written as \( y = mx + b \. \) Here, \( m \) represents the slope, and \( b \) denotes the y-intercept. The slope \( m \) is a measure of how steep the line is. It indicates how much the \( y \) value changes for a change in \( x \). For example, if \( m \) is 2, it means that for every unit increase in \( x \), \( y \) increases by 2 units. The y-intercept, \( b \), is the point where the line crosses the y-axis. This is where \( x = 0\. \) Together, \( m \) and \( b \) give us a clear picture of the line's behavior and position on a graph.

Y-Intercept

The y-intercept is an important feature of any linear equation in the slope-intercept form \( (y = mx + b)\). It describes the point at which the line crosses the y-axis. This is where the value of \( x \) is zero. To determine the y-intercept from a linear equation, simply look at the \( b \) value in the equation. For instance, in the equation \( y = -x - 1 \), the y-intercept \( b \) is -1. This means that the line crosses the y-axis at the point \( (0, -1)\). This is useful because it gives us an anchor point from which we can plot the rest of the line.

Finding Equation of a Line

Let's tackle the problem of finding the equation of a line. Suppose a line passes through the point \( (-3, 2) \) and has a slope of -1. We use the slope-intercept form \( y = mx + b \) to find the equation. First, substitute the slope \( -1 \) into the equation: ewline ewline \( y = -1x + b \), simplify it to \( y = -x + b \). ewline ewline Next, to find the y-intercept \( b \), use the given point \( (-3, 2) \). When \( x = -3 \), \( y = 2 \). Substitute these values into the equation: \( 2 = -(-3) + b \). Simplifying this results in \( 2 = 3 + b \), and further solving gives us \( b = -1 \). ewline ewline Finally, substitute \( b = -1 \) back into the equation: \( y = -x - 1\). Now, we have the line equation that satisfies the given conditions.