Problem 22 - Precalculus Mathematics for Calculus

Question

Find an equation of the line that satisfies the given conditions. Through \((-3,-5) ; \quad\) 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: Understand the Point-Slope Form

The point-slope form of the equation of a line is given by \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \( m \) is the slope.

Step 2: Identify Given Values

We know that the point \((-3, -5)\) is on the line, so \(x_1 = -3\) and \(y_1 = -5\). The slope \(m\) is given as \(-\frac{7}{2}\).

Step 3: Substitute into Point-Slope Form

Substitute \(x_1 = -3\), \(y_1 = -5\), and \(m = -\frac{7}{2}\) into the point-slope form:\[ y + 5 = -\frac{7}{2}(x + 3) \]

Step 4: Simplify the Equation

Distribute \(-\frac{7}{2}\) through the \(x + 3\):\( y + 5 = -\frac{7}{2}x -\frac{21}{2} \).Next, subtract 5 from both sides to simplify:\( y = -\frac{7}{2}x -\frac{21}{2} - 5 \).Convert 5 into fraction form \(\frac{10}{2}\):\( y = -\frac{7}{2}x -\frac{21}{2} - \frac{10}{2} \).Combine like terms to get:\( y = -\frac{7}{2}x -\frac{31}{2} \).

Step 5: Finalize the Equation

The equation of the line in slope-intercept form is:\( y = -\frac{7}{2}x -\frac{31}{2} \).This represents the equation of the line that goes through the point \((-3, -5)\) with a slope of \(-\frac{7}{2}\).

Key Concepts

Point-Slope Form

The point-slope form is one of the ways to express the equation of a line.
It is very useful when you know a point on the line and its slope. The formula is expressed as:

• \( y - y_1 = m(x - x_1) \)

In this formula, \((x_1, y_1)\) represents a specific point on the line.
The variable \(m\) is the slope of the line. This form is particularly handy for writing the equation quickly using a point and the slope without needing any further transformation.
This makes it easier to visualize the line's behavior around the known point.

Slope-Intercept Form

The slope-intercept form is maybe the most commonly used form to represent linear equations.
It is written as \( y = mx + b \). In this case:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, where the line crosses the y-axis.

This form is convenient because it directly shows the slope and intercept, making it easier to graph.
Transforming from point-slope to slope-intercept usually involves algebraic manipulation to simplify the equation and find \(b\).
For the line passing through point \((-3, -5)\) with a slope of \(-\frac{7}{2}\), the slope-intercept form is \( y = -\frac{7}{2}x -\frac{31}{2} \). This is derived by isolating \(y\) in the point-slope form.

Linear Equations

Linear equations are equations of the first order that do not have variables with exponents higher than one.
In a two-dimensional space, they graph as straight lines. The standard forms of these equations include the slope-intercept and point-slope forms.
Linear equations can describe relationships where something changes at a constant rate. For example:

• A car traveling at a constant speed.
• Constant growth or decay phenomena.

They serve as building blocks in many mathematical applications, helping us understand and predict behavior in a straightforward manner.
In coordinate geometry, understanding linear equations is crucial to solving more complex geometric problems.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, integrates algebra with geometry.
It is primarily concerned with the coordinates (x, y) used to identify points on a plane. This branch allows us to use equations to understand geometric principles.
In relation to lines, coordinate geometry helps in analyzing and understanding:

• Position and distance between points.
• How lines are situated in space, using forms like point-slope and slope-intercept.
• Solving geometric problems by assigning values and finding relationships.

Through coordinate geometry, equations like that of a line provide a concrete way to predict and assess different aspects such as slopes, intersections, and more in a spatial context.