Problem 18 - Algebra and Trigonometry

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 is \( y = -\frac{7}{2}x - \frac{31}{2} \).

Step 1: Understand the Formula

The equation of a line in slope-intercept form is given by \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept. We are provided with a point and a slope, and we will utilize these to find the equation.

Step 2: Use Point-Slope Form

The point-slope form of the equation of a line is \( y - y_1 = m(x - x_1) \). Given point \((-3, -5)\) and slope \( m = -\frac{7}{2} \), substitute these values into the formula: \( y - (-5) = -\frac{7}{2}(x - (-3)) \). This simplifies to \( y + 5 = -\frac{7}{2}(x + 3) \).

Step 3: Simplify the Equation

Distribute \(-\frac{7}{2}\) through the terms in the parentheses: \( y + 5 = -\frac{7}{2}x - \frac{21}{2} \).

Step 4: Solve for y

To convert to slope-intercept form, solve for \( y \) by subtracting 5 from both sides: \( y = -\frac{7}{2}x - \frac{21}{2} - 5 \).

Step 5: Simplify Further

Combine the constants on the right side: \( -\frac{21}{2} - 5 \) can be rewritten as \( -\frac{21}{2} - \frac{10}{2} = -\frac{31}{2} \). This provides us the final equation, \( y = -\frac{7}{2}x - \frac{31}{2} \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a widely used representation because it easily shows the slope and y-intercept of a line. This form is set up as \( y = mx + b \), where:

• \( m \) is the slope of the line, indicating its steepness and direction.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

The beauty of this form is its simplicity and intuitive grasp. By knowing \( m \) and \( b \), one can quickly visualize the line's behavior on a graph.
To transform an equation from point-slope to slope-intercept form, we solve for \( y \). After inserting the given slope and calculating the intercept, a clear linear equation emerges, ready for graphing or further analysis.

Point-Slope Form

When you're given a point and a slope, the point-slope form provides a straightforward way to construct the equation of a line. It is expressed as \( y - y_1 = m(x - x_1) \), where:

• \((x_1, y_1)\) are the coordinates of a specific point the line passes through.
• \( m \) is, as always, the slope.

This form is particularly handy when dealing with problems where a specific point on the line is known. It ties the slope to actual coordinates, making it easier to visualize how the line angles through a particular point. Once the equation is set, further simplifications can convert it into slope-intercept form, which is often more convenient for plotting or solving additional problems.

Linear Equations

Linear equations describe a straight line on a plane and are foundational in algebra. They typically combine constants and variables, forming equations that represent a direct relationship between two quantities.
In their basic forms, such as the slope-intercept (\( y = mx + b \)) and point-slope (\( y - y_1 = m(x - x_1) \)), linear equations provide vital information about the geometric entities they describe.

• They show the relationship between two variables, often depicting real-world problems like speed versus time or cost versus quantity.
• Understanding how to manipulate these equations, whether by changing forms or solving for specific variables, is crucial for applying algebra effectively.

By learning to interpret and work with linear equations, one gains a powerful toolset for tackling a wide array of mathematical challenges.