Problem 52 - College Algebra

Question

Write an equation for a line parallel to \(f(x)=-5 x-3\) and passing through the point \((2,-12)\)

Step-by-Step Solution

Answer

The equation is \( y = -5x - 2 \).

Step 1: Understanding the Slope of Parallel Lines

To find an equation of a line parallel to a given line, you need to use the same slope. The given line is \( f(x) = -5x - 3 \). This line is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. Therefore, the slope of our line is \( -5 \). Parallel lines share the same slope.

Step 2: Using the Point and Slope to Find the Equation

Now that we have the slope \( m = -5 \), we use the point-slope form \( y - y_1 = m(x - x_1) \) with the point \( (2, -12) \). Substitute \( m = -5 \), \( x_1 = 2 \), and \( y_1 = -12 \):\[y + 12 = -5(x - 2)\]

Step 3: Simplifying the Equation

Distribute the \(-5\) and simplify the equation:\[y + 12 = -5x + 10\]Subtract 12 from both sides to solve for \( y \):\[y = -5x - 2\]

Step 4: Final Equation

The equation of the line parallel to \( f(x) = -5x - 3 \) and passing through the point \((2, -12)\) is \( y = -5x - 2 \).

Key Concepts

Equation of a Line

An equation of a line is an essential mathematical expression representing a straight line on the Cartesian plane. It defines the entirety of the line, displaying all its points. There are different forms of expressing this equation. Each form reveals different properties of the line:

• The most common forms we will encounter include slope-intercept form and point-slope form.
• The equation helps us understand crucial details such as slope and intercepts.
• It is a foundational concept in algebra and geometry, useful for solving practical real-world problems.

Choosing the most suitable form depends on the given information about the line. Understanding each form's nuances aids in selecting and efficiently transitioning between them.

Slope-Intercept Form

The slope-intercept form of a line's equation is particularly user-friendly. It's expressed as \( y = mx + b \). In this format:

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

This form quickly reveals the line's slope and y-intercept, making it useful for graphing a line from two simple parameters. For example, the line \( f(x) = -5x - 3 \) has a slope of \(-5\) and a y-intercept at \(-3\). This immediate understanding makes it easy to compare and graph lines, particularly in tasks involving parallel lines.

Point-Slope Form

The point-slope form of a line's equation is useful when we know a point on the line and its slope. It is expressed as \( y - y_1 = m(x - x_1) \). Here's how it works:

• \( m \) denotes the slope of the line.
• \((x_1, y_1)\) is a known coordinate point on the line.

Ideal for constructing an equation when point and slope data is given, this form makes it straightforward to plug in values and derive an equation. For example, with a slope of \(-5\) and passing through \((2, -12)\), we find the equation to be \( y + 12 = -5(x - 2) \), which further simplifies as needed.

Slope of a Line

The slope () of a line is a measure of its inclination or steepness. It describes how much the function changes vertically for a change in the horizontal direction. The formula for calculating slope when given two points, \((x_1, y_1)\) and \((x_2, y_2)\), is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

The slope reveals whether a line rises (positive slope), falls (negative slope), or is horizontal (zero slope). Lines that share the same slope are parallel; thus, they never intersect. For example, in our problem, the slope \(-5\) implies a downward trajectory, critical for finding a parallel line.