Problem 52 - College Algebra with Corequisite Support

Question

For the following exercises, write an equation for the line described. 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: Identify Slope of Given Line

The given line equation is in the form \(f(x) = -5x - 3\). The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope. Therefore, the slope (\(m\)) of the given line is \(-5\).

Step 2: Understand Parallel Line Slope

Two lines are parallel if and only if they have the same slope. Therefore, the slope of the line we need to find is also \(-5\).

Step 3: Apply Point-Slope Form

We use the point-slope form equation, which is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Substitute \((-5\)) for \(m\) and the point \((2, -12)\) for \((x_1, y_1)\).

Step 4: Substitute and Simplify

Plug the point and slope into the point-slope form: \(y - (-12) = -5(x - 2)\). Simplify this equation to get: \(y + 12 = -5x + 10\), then subtract 12 from both sides: \(y = -5x - 2\).

Step 5: Outcome

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

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is a very convenient way to write the equation of a line. It looks like this: \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. This form is particularly useful because it clearly shows both the slope and the y-intercept, making it easy to graph the line or understand its behavior.

• **Slope (\( m \)):** The slope indicates the steepness of the line, or how much \( y \) changes for a one-unit change in \( x \). In our example, the slope is \(-5\), meaning the line falls by 5 units for every 1 unit move to the right.
• **Y-intercept (\( b \)):** The y-intercept tells you where the line crosses the y-axis. For the given equation \( f(x) = -5x - 3 \), the y-intercept is \(-3\).

This form is ideal for understanding the fundamental properties of a line in a quick glance. Knowing the slope and y-intercept makes it easy to identify parallelism, which is crucial when finding parallel lines as in the exercise.

Point-Slope Form

The point-slope form is another way to express the equation of a line. It comes in handy when you know the slope and a particular point that the line passes through. The formula is: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a specific point on the line.

This form is especially useful when given one point on the line and the slope, as is the case with finding a line parallel to another. Since parallel lines share the same slope, you take the slope from the given line (\(-5\) in this case) and the point the new line must pass through (here \((2, -12)\)).

• **Why Use Point-Slope Form:** It allows for expressing a line using just its slope and one point, making it versatile and useful for this type of problem.
• **Executing the Formula:** Plug \(-5\) in for \( m \), \( 2 \) for \( x_1 \), and \(-12\) for \( y_1 \), leading to the equation: \( y + 12 = -5(x - 2) \). Simplify this to find the line in its more comprehensive form.

This method lets you transition from knowing partial details about a line to formulating a complete line equation.

Equation of a Line

Understanding the equation of a line is crucial in graphing and solving geometric problems. For instance, once you have a line's slope and point, you can establish its entire equation in several forms, such as the slope-intercept or point-slope forms.

Constructing the equation involves using the given information efficiently.

• For parallel lines: These have identical slopes, which is the key to solving the problem at hand.
• The point-slope form provides the means to tie that slope to a specific point to create an equation that holds true for the entire line.

Once the new line's equation is formed, as in \( y = -5x - 2 \), we can analyze or graph it and compare it to the original line \( f(x) = -5x - 3 \) to see how they align parallelly across the plane.

The ability to transition between forms as needed aids in understanding and presenting lines in the most beneficial way.