Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-32: Find a general form of an equation of the line through the point \(A\) that satisfies the given condition. $$ A(7,-3) ; \quad \text { perpendicular to the line } 2 x-5 y=8 $$

Step-by-Step Solution

Answer

The equation is \(5x + 2y - 29 = 0\).

Step 1: Determine the slope of the given line

First, identify the slope of the line given by the equation \(2x - 5y = 8\). Rewrite the equation in slope-intercept form, \(y = mx + b\), to find its slope. Begin by isolating \(y\) on one side:\[2x - 5y = 8\]Subtract \(2x\) from both sides:\[-5y = -2x + 8\]Divide everything by \(-5\):\[y = \frac{2}{5}x - \frac{8}{5}\]Thus, the slope \(m\) of the given line is \(\frac{2}{5}\).

Step 2: Find the slope of the perpendicular line

The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. The slope of the given line is \(\frac{2}{5}\), so the negative reciprocal is \(-\frac{5}{2}\). This is the slope of the line we need to find.

Step 3: Use the point-slope form for the line

Use the point-slope form equation \(y - y_1 = m(x - x_1)\) to write the equation of the line with slope \(-\frac{5}{2}\) passing through the point \(A(7, -3)\). Here, \(x_1 = 7\), \(y_1 = -3\), and \(m = -\frac{5}{2}\):\[y + 3 = -\frac{5}{2}(x - 7)\]

Step 4: Convert to general form

Simplify the equation to general form, \(Ax + By + C = 0\). Start with the point-slope form:\[y + 3 = -\frac{5}{2}(x - 7)\]Distribute the \(-\frac{5}{2}\):\[y + 3 = -\frac{5}{2}x + \frac{35}{2}\]Multiply everything by 2 to eliminate the fraction:\[2y + 6 = -5x + 35\]Rearrange to get the general form:\[5x + 2y - 29 = 0\]Thus, the general form of the equation is \(5x + 2y - 29 = 0\).

Key Concepts

Point-slope form

When we talk about the point-slope form of a linear equation, we're referring to a specific format used to define a straight line. This is especially useful when we have a point on the line and the slope of the line. The point-slope equation is:\[ y - y_1 = m(x - x_1) \]Where:

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

In the exercise provided, we transformed our equation by inserting the negative reciprocal slope of \(-\frac{5}{2}\), which is perpendicular to the slope \(\frac{2}{5}\) of the given line, and the point \( A(7, -3) \). This allows us to produce an equation in point-slope form:\[ y + 3 = -\frac{5}{2}(x - 7) \]This form is convenient for many applications because it directly uses a point on the line, reflecting situations where you don't need to compute additional intercepts as you do in other forms.

Slope-intercept form

The slope-intercept form is another way of expressing the equation of a line. It's popular in algebra because it provides immediate insight into the slope and y-intercept of the line. The slope-intercept form is:\[ y = mx + b \]Here, \( m \) represents the slope, and \( b \) is the y-intercept where the line crosses the y-axis. To transform from point-slope to slope-intercept form:

• First, solve for \( y \) in terms of \( x \).
• Simplify the resulting expression.

In the exercise, the given line was rewritten in this form to identify the slope:\[ y = \frac{2}{5}x - \frac{8}{5} \]By transforming to slope-intercept form, it becomes evident that the slope is \(\frac{2}{5}\) and the y-intercept is \(-\frac{8}{5}\). For the line we created, direct conversion isn't necessary, but understanding this format helps in visualizing and sketching graphs.

General form of a line equation

The general form of a line equation is another commonly used representation. This standard format is expressed as:\[ Ax + By + C = 0 \]Where \( A, B, \) and \( C \) are integers, and typically \( A \) should be a non-negative integer. The advantage of this form is its unified representation of all linear equations, even vertical lines, which cannot be expressed in slope-intercept form.For the exercise, after using point-slope form, we converted the equation to the general form with these steps:

• Begin with the point-slope equation.
• Clear fractions and rearrange terms.

This process led us to the general form:\[ 5x + 2y - 29 = 0 \]This form is useful for analyzing and comparing different lines, as it provides a straightforward way to discern intersection points and parallelism by examining the coefficients.