Problem 22 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((-3,-5) ;\) slope \(-\frac{7}{2}\)

Step-by-Step Solution

Answer

The equation of the line is \(y = -\frac{7}{2}x - \frac{31}{2}\).

Step 1: Understanding the Problem

We need to find an equation of the line that passes through a given point \((-3, -5)\) with a given slope \(-\frac{7}{2}\). The goal is to express the equation in the slope-intercept form, \(y = mx + b\).

Step 2: Substitute the Slope into the Equation

Given the slope \(m = -\frac{7}{2}\), we substitute this value into the slope-intercept form of the equation: \(y = -\frac{7}{2}x + b\).

Step 3: Use the Point to Find the Y-Intercept

To find the y-intercept \(b\), use the given point \((-3, -5)\). Substitute \(x = -3\) and \(y = -5\) into the equation: \(-5 = -\frac{7}{2}(-3) + b\).

Step 4: Calculate the Y-Intercept

Simplify the expression: \(-5 = \frac{21}{2} + b\). Solve for \(b\) by subtracting \(\frac{21}{2}\) from both sides to get \(b = -5 - \frac{21}{2}\). We need a common denominator, converting \(-5\) to \(-\frac{10}{2}\), thus \(b = -\frac{10}{2} - \frac{21}{2} = -\frac{31}{2}\).

Step 5: Write the Final Equation

Now that we have \(m = -\frac{7}{2}\) and \(b = -\frac{31}{2}\), we can write the equation of the line as \(y = -\frac{7}{2}x - \frac{31}{2}\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a straight line. It's expressed as \(y = mx + b\). This form is popular because it immediately gives you two key pieces of information about the line:

• The slope \(m\)
• The y-intercept \(b\)

Using the slope-intercept form is like having a map to understand how a line behaves.
The slope \(m\) shows how steep the line is. It tells you how much \(y\) changes for a one-unit change in \(x\). A positive slope means the line goes upwards; a negative slope means it goes downwards as you move right.
The y-intercept \(b\) is where the line crosses the y-axis. This point is really helpful because it anchors your line on the graph.

Equation of a Line

To find the equation of a line, you need at least two pieces of information: one point on the line and the slope of the line. Once you have these, you can use the slope-intercept form \(y = mx + b\) to build the equation.
In our given problem, you know that the line passes through the point \((-3, -5)\) and has a slope of \(-\frac{7}{2}\). This means:

• The line is quite steep because the slope is \(-3.5\) (the negative sign shows it descends).
• Your task is to determine the y-intercept \(b\) to complete the equation.

Insert the slope into the equation and use the point to solve for \(b\). Plugging the coordinates of the known point into the slope-intercept equation lets us solve for the missing y-intercept and complete the picture.

Finding Y-Intercept

Finding the y-intercept \(b\) is like finding the last piece of a puzzle to complete your line's equation. From this foundational equation \(y = mx + b\), the goal is to isolate \(b\).
Once you place the slope you have, \(-\frac{7}{2}\), into the equation, you'll start solving for \(b\) by inserting the point coordinates. The given point \((-3, -5)\) provides the \(x\) and \(y\) values to substitute into the equation.

• Substitution gives: \(-5 = -\frac{7}{2}(-3) + b\)
• Simplify to calculate \(b\)
• Finding \(b\) involves basic algebra: from simplifying fractions to obtaining a shared denominator.

After computation, our y-intercept \(b\) is \(-\frac{31}{2}\). With both \(m\) and \(b\), the line's equation is now clear: \(y = -\frac{7}{2}x - \frac{31}{2}\). Now, you have a complete understanding of this line's equation.