Problem 46 - Elementary Algebra

Question

Write the equation of the line that satisfies the given conditions. Express final equations in standard form. \(x\) intercept of \(-1\) and \(y\) intercept of \(-3\)

Step-by-Step Solution

Answer

The standard form equation is \(3x + y = -3\).

Step 1: Understanding the Intercepts

The x-intercept of a line is the point where the line crosses the x-axis. In this case, it is at \((-1, 0)\). The y-intercept is where the line crosses the y-axis, which here is at \((0, -3)\).

Step 2: Slope Calculation

To find the slope of the line, we use the two intercept points \((-1, 0)\) and \((0, -3)\). The slope \(m\) is calculated as: \[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 0}{0 - (-1)} = \frac{-3}{1} = -3\]

Step 3: Equation in Slope-Intercept Form

Using the slope \(m = -3\) and the y-intercept \((0, -3)\), we can write the equation in slope-intercept form (\(y = mx + b\)): \[y = -3x - 3\]

Step 4: Convert to Standard Form

The standard form of a line's equation is \(Ax + By = C\). To convert \(y = -3x - 3\) to standard form, rearrange the equation: \[3x + y = -3\]

Key Concepts

Understanding the x-intercept

An x-intercept occurs where a line crosses the x-axis. At this point, the y-value is always zero because the line is on the horizontal axis. In our problem, the x-intercept is given as \((-1, 0)\).
This means the line intersects the x-axis at \(x = -1\). Finding the x-intercept is important as it provides crucial information about the orientation of the line.
Why is this valuable? It helps us establish one of the two essential points needed to calculate the slope of the line.

Identifying the y-intercept

The y-intercept is similar to the x-intercept but occurs where the line crosses the y-axis. At this point, the x-value is zero, because the line is on the vertical axis. For our equation, the y-intercept is found at the point \((0, -3)\).
This indicates that the line crosses the y-axis at \(y = -3\). Understanding the y-intercept is crucial because it directly informs us of the vertical placement of the line. It also provides the starting point when the equation is written in slope-intercept form \(y = mx + b\), where \(b\) represents the y-intercept.

Slope Calculation Basics

Calculating the slope of a line involves determining how much and in which direction the line rises or falls. Slope (\(m\)) is found using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
For our points \((-1, 0)\) and \((0, -3)\), the slope is calculated as:

• You subtract the y-values: \((-3) - 0 = -3\)
• You subtract the x-values: \(0 - (-1) = 1\)
• The slope equation becomes: \(\frac{-3}{1} = -3\)

The negative value of the slope indicates the line moves downwards from left to right.

Converting to Standard Form

Equations of a line can be presented in various forms, and one of them is the standard form: \(Ax + By = C\). This is useful for mathematical analysis and graphing. Given our slope-intercept form \(y = -3x - 3\), we can convert it.
To achieve this,

• We need to get all variables and constants on one side of the equation with x and y coefficients being integers.
• Reorder and transpose: \(3x + y = -3\)

This achieves the desired \(Ax + By = C\) format and helps communicate the equation clearly.