Problem 34 - Intermediate Algebra

Question

\(x\) intercept of \(-1\) and \(y\) intercept of \(-3\)

Step-by-Step Solution

Answer

The line's equation is \(y = -3x - 3\).

Step 1: Identify the Objective

We need to find the equation of a line that has an x-intercept of -1 and a y-intercept of -3. The line's equation will be linear, in the form of \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

Step 2: Determine Intercepts

The x-intercept means the line crosses the x-axis at \((-1, 0)\). The y-intercept means the line crosses the y-axis at \((0, -3)\). These points provide the values needed to find the slope.

Step 3: Calculate the Slope

The slope \(m\) is calculated using the formula for slope: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the intercept points \((x_1, y_1) = (-1, 0)\) and \((x_2, y_2) = (0, -3)\), we find \[m = \frac{-3 - 0}{0 - (-1)} = \frac{-3}{1} = -3.\]

Step 4: Formulate the Equation

With the slope \(m = -3\) and y-intercept \(c = -3\), insert these into the slope-intercept form \(y = mx + c\) to get the equation \(y = -3x - 3\).

Key Concepts

Understanding the X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of y is zero. In simpler terms, it's where the line "touches" the x-axis. In our exercise, the x-intercept is given as -1. This means the line crosses the x-axis at the point \((-1, 0)\).

• To find an x-intercept, set \(y = 0\) in the equation of the line.
• Solve for \(x\) to find the specific x-intercept.

Identifying the x-intercept is crucial because it helps in determining the slope of the line when paired with the y-intercept.

Exploring the Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. Here, the value of x is zero. For our specific problem, the y-intercept is -3, meaning the line crosses the y-axis at the point \((0, -3)\).

• The y-intercept is often denoted by \(c\) in the slope-intercept form of a line.
• When the line's equation is given, you can find the y-intercept by setting \(x = 0\) and solving for \(y\).

Knowing the y-intercept is essential for constructing the line's equation in slope-intercept form.

Decoding the Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \(y = mx + c\). This form is popular because it provides straightforward information about the line's slope and y-intercept. In this form:

• \(m\) represents the slope of the line.
• \(c\) is the y-intercept.

This form makes it easy to graph the line and understand its direction and steepness. For example, in our problem, we derived the equation \(y = -3x - 3\). Here:

• The slope \(m = -3\) indicates that the line falls downwards as you move from left to right.
• The y-intercept \(c = -3\) tells us the starting point of the line on the y-axis.

Understanding the slope-intercept form is a key tool for analyzing linear equations.

Mastering Slope Calculation

The slope of a line measures its steepness and direction. It is calculated using two points on the line. For our exercise, the relevant points are the intercepts \((-1, 0)\) and \((0, -3)\).

• The slope \(m\) is calculated by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
• Using our points, the calculation becomes \(m = \frac{-3 - 0}{0 - (-1)} = \frac{-3}{1} = -3\).

A negative slope, like -3, implies the line is falling as it moves along the x-axis. Calculating the slope is pivotal for forming the equation of the line in slope-intercept form and understanding the line's characteristics.