Problem 22 - Algebra for College Students

Question

Determine the linear function whose graph is a line that is parallel to the line \(g(x)=-3 x-4\) and contains the point \((2,7) . \quad f(x)=-3 x+13\)

Step-by-Step Solution

Answer

The linear function is \( f(x) = -3x + 13 \).

Step 1: Identify the Slope of Line g(x)

The function you are given, \( g(x) = -3x - 4 \), is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. Thus, the slope of \( g(x) \) is \( -3 \). Parallel lines have the same slope.

Step 2: Use Point-Slope Form for Line f(x)

Since the slope of the new line \( f(x) \) must be the same as \( g(x) \), the slope \( m = -3 \). The point-slope form of the equation of a line is \( y - y_1 = m(x - x_1) \). Plug in \( m = -3 \) and the point \( (2, 7) \): \[ y - 7 = -3(x - 2) \]

Step 3: Simplify to Slope-Intercept Form

Distribute and simplify the equation: \[ y - 7 = -3x + 6 \]Add 7 to both sides to solve for \( y \):\[ y = -3x + 13 \] Thus, the equation of the line \( f(x) \) is \( f(x) = -3x + 13 \).

Key Concepts

Parallel Lines

When given a line, it's interesting to consider other lines that are parallel to it. Parallel lines are lines in the same plane that never intersect. They have the same direction, meaning they must have the same slope. This is a key property in coordinate geometry. For example, if you have a line with an equation in the slope-intercept form such as \( g(x) = -3x - 4 \), the slope of this line is \(-3\). A new line parallel to this one must also have a slope of \(-3\). However, the parallel line can have a different y-intercept, which allows it to be entirely distinct in its position relative to the y-axis.

Here are the main takeaways about parallel lines:

• Parallel lines have identical slopes.
• They do not meet at any point.
• The parallel relationship is maintained across different y-intercepts, affecting their position but not their direction.

This understanding is instrumental when dealing with linear equations, especially in problems that involve finding parallel lines through given points.

Point-Slope Form

The point-slope form of a linear equation is incredibly useful, especially when you know a line's slope and a point through which it passes. The form is expressed as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a point on the line. This format allows you to quickly determine the equation of a line if you know these two pieces of information.

In the exercise, we wanted a line parallel to \( g(x) = -3x - 4 \) that goes through the point \((2,7)\). We identified the slope \( m = -3 \) and applied it to the point-slope form. By substituting \( x_1 = 2 \) and \( y_1 = 7 \), we get:

• Equation: \( y - 7 = -3(x - 2) \)
• This equation directly uses the known slope and point.

This approach simplifies finding the equation of any line when given a point and a slope, especially when a line must be parallel or perpendicular to another.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most straightforward and popular ways to express a line. It has the format \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form is helpful because it immediately shows the line's slope and where it crosses the y-axis. In our problem, after finding the equation in point-slope form, simplifying it leads us back to the slope-intercept form.

For the line \( f(x) \) with a slope of \(-3\) that passes through \((2, 7)\), we converted it to slope-intercept form:

• Starting from: \( y - 7 = -3(x - 2) \)
• Distribute: \( y - 7 = -3x + 6 \)
• Add 7 to both sides: \( y = -3x + 13 \)

This final form \( f(x) = -3x + 13 \) makes it easy to visualize and understand the line's characteristics, such as inclination and where it sits on the y-axis. The simplicity of reading both the slope and intercept makes this form ideal for graphing linear functions.