Problem 53 - College Algebra

Question

Write an equation for a line parallel to \(g(x)=3 x-1\) and passing through the point \((4,9) .\)

Step-by-Step Solution

Answer

The equation is \(y = 3x - 3\).

Step 1: Identify the Slope of the Line

The equation given is \(g(x) = 3x - 1\). This is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope of the line. Hence, the slope of line \(g(x)\) is 3.

Step 2: Determine the Slope of the Parallel Line

Lines that are parallel have the same slope. Therefore, the slope of the line we want to find is also 3.

Step 3: Use the Point-Slope Form

To find the equation of the line, use the point-slope form \(y - y_1 = m(x - x_1)\). Here \((x_1, y_1) = (4, 9)\) and \(m = 3\). Substituting these values, we get: \(y - 9 = 3(x - 4)\).

Step 4: Simplify the Equation

Simplify the equation \(y - 9 = 3(x - 4)\). First, distribute the slope: \(y - 9 = 3x - 12\). Then, add 9 to both sides to isolate \(y\): \(y = 3x - 3\).

Step 5: Write the Final Equation

The simplified equation of the line parallel to \(g(x) = 3x - 1\) and passing through \((4, 9)\) is \(y = 3x - 3\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to write the equation of a line. This form is expressed as \(y = mx + b\), where:

• \(m\) is the slope of the line. It represents how steep the line is or how much \(y\) increases for each increase in \(x\).
• \(b\) is the y-intercept. It indicates where the line crosses the y-axis.

To understand the slope, remember it's like the incline of a ramp. A larger \(m\) means a steeper incline. In the given exercise, the line \(g(x) = 3x - 1\) helps us identify that \(m = 3\). This means for every 1 unit where \(x\) increases, \(y\) increases by 3 units.
When writing an equation for any line, identifying the slope and y-intercept using the slope-intercept form makes analyzing and graphing lines much simpler. For parallel lines, you only need to ensure that the slope \(m\) is the same.

Point-Slope Form

The point-slope form of a line provides a way to write an equation when you know a single point and the slope. It's given by the expression \(y - y_1 = m(x - x_1)\), where:

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

Point-slope form is especially useful when a line passes through a specific point and has a defined slope. In our example, with a parallel line to \(g(x) = 3x - 1\) going through the point \((4, 9)\), the slope \(m = 3\) and the point \((4, 9)\) are crucial.
By substituting these values into the point-slope form: \(y - 9 = 3(x - 4)\), we derive an equation containing all necessary pieces of information. This form allows easy transformation into the slope-intercept form for graphing or further analysis.

Coordinate Geometry

Coordinate geometry, or analytic geometry, helps us visualize algebraic equations by transforming them into geometric shapes. The fundamental idea is to represent algebraic equations as points or lines on a plane.

• The x-axis and y-axis create a coordinate plane where every point is defined by an \((x, y)\) pair.
• Lines on this plane can be described using linear equations.

Understanding coordinate geometry enables you to plot lines using equations like the slope-intercept form \(y = mx + b\) or point-slope form \(y - y_1 = m(x - x_1)\).
In the given exercise, by converting the information into an equation, we can visualize a line parallel to \(g(x)\) passing through the specific point \((4, 9)\). By bridging algebra and geometry, you get a more holistic view of how mathematical concepts represent real-world objects.