Problem 78 - Algebra A Combined Function

Question

Write an equation in standard form of the line that contains the point (4,0) and is perpendicular to the line \(y=-2 x+3\)

Step-by-Step Solution

Answer

The standard form of the line is \(x - 2y = 4\).

Step 1: Understand the Slope of the Given Line

The given line is represented by the equation \(y = -2x + 3\). This equation is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope. Thus, the slope of the given line \(m_1 = -2\).

Step 2: Define the Slope of the Perpendicular Line

For a line to be perpendicular to another, the product of their slopes must equal \(-1\). If the slope of the given line is \(m_1 = -2\), then the slope \(m_2\) of the line perpendicular to it is the negative reciprocal. So, \(m_2 = \frac{1}{2}\).

Step 3: Formulate the Point-Slope Equation

We have a point \((4,0)\) and a slope \(\frac{1}{2}\). Using the point-slope form \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line, we get: \(y - 0 = \frac{1}{2}(x - 4)\).

Step 4: Simplify to Slope-Intercept Form

Expand the point-slope equation: \(y = \frac{1}{2}x - 2\). This equation is a simplified version in slope-intercept form \(y = mx + b\).

Step 5: Convert to Standard Form

The standard form of a line is \(Ax + By = C\). Rearrange the equation \(y = \frac{1}{2}x - 2\) by getting rid of the fraction and collecting terms: Multiply the entire equation by 2 to eliminate the fraction: \(2y = x - 4\). Rearranging, we get: \(-x + 2y = -4\). Multiply through by -1 for standard convention: \(x - 2y = 4\).

Key Concepts

Slope-Intercept Form

Let's start understanding lines by diving into the slope-intercept form, which is one of the most intuitive ways to express a linear equation. The general formula is given by \(y = mx + b\). In this equation:

• \(m\) stands for the slope of the line, representing how steep the line is.
• \(b\) is the y-intercept, the point where the line intersects the y-axis.

The beauty of this form is its simplicity. As soon as you recognize a line in this format, you instantly know both its slope and where it crosses the vertical axis.
For instance, in the line \(y = -2x + 3\), the slope \(m = -2\) and the y-intercept \(b = 3\). This lets you visualize that for every unit increase in \(x\), \(y\) decreases by 2, and the line cuts through the y-axis at the point (0,3). This straightforward depiction can help in quickly sketching lines and understanding their behavior.

Point-Slope Form

The point-slope form is incredibly useful when you know a point on the line and the slope. The formula here is \(y - y_1 = m(x - x_1)\), where:

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

This form is particularly handy when you have data points or for lines passing through specific coordinates.
For example, if you have a line passing through the point (4,0) with a slope of \(\frac{1}{2}\), the point-slope form becomes \(y - 0 = \frac{1}{2}(x - 4)\). From this equation, you see how the line behaves around that specific point, and it allows you to anchor the slope around known coordinates. It's an excellent stepping stone towards converting the line into other forms, such as slope-intercept or standard.

Standard Form of a Line

The standard form of a line is another way to describe linear equations, expressed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) should typically be non-negative. Unlike the slope-intercept form, the standard form is not easily recognizable for slope or intercept, but it's very useful in certain situations.
This format is very structured and is often preferred for eliminating fractions and maintaining integer coefficients in equations.

• It's handy for solving systems of equations or when working with integer-based coefficients.
• You can arrange any linear equation into the standard format, provided you adjust the terms to fit \(Ax + By = C\).

In the original problem, we took the equation \(y = \frac{1}{2}x - 2\), removed the fraction by multiplying through by 2, and rearranged it to get \(x - 2y = 4\). This conversion highlights the versatility of manipulating lines for different calculation needs while maintaining the line's intrinsic geometric properties.