Problem 31 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((-1,-2) ;\) perpendicular to the line \(2 x+5 y+8=0\)

Step-by-Step Solution

Answer

The equation is \(y = \frac{5}{2}x + \frac{1}{2}\).

Step 1: Identify the slope of the given line

The equation of the line is given as \(2x + 5y + 8 = 0\). We write this in slope-intercept form, \(y = mx + b\), to find its slope \(m\). Start by isolating \(y\): \(5y = -2x - 8\). Dividing each term by 5 gives \(y = -\frac{2}{5}x - \frac{8}{5}\). Thus, the slope \(m\) of the given line is \(-\frac{2}{5}\).

Step 2: Determine the slope of the line perpendicular to the given line

To find the slope of the line perpendicular to the given line, we take the negative reciprocal of \(-\frac{2}{5}\). The negative reciprocal is \(\frac{5}{2}\). Thus, the slope of the perpendicular line is \(\frac{5}{2}\).

Step 3: Use the point-slope form to find the equation of the line

We have a point \((-1, -2)\) and a slope \(\frac{5}{2}\). We use the point-slope formula: \(y - y_1 = m(x - x_1)\). Plugging in the values, we have \(y - (-2) = \frac{5}{2}(x - (-1))\), which simplifies to \(y + 2 = \frac{5}{2}(x + 1)\).

Step 4: Simplify the equation into slope-intercept form

Distribute \(\frac{5}{2}\) through \(x + 1\) to get \(y + 2 = \frac{5}{2}x + \frac{5}{2}\). Subtract 2 from both sides to isolate \(y\): \(y = \frac{5}{2}x + \frac{5}{2} - 2\). Simplify to get \(y = \frac{5}{2}x + \frac{1}{2}\), which is the slope-intercept form of the desired line.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express the equation of a line. It is given by the formula \(y = mx + b\), where:

• \(m\) represents the slope of the line,
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

To find the slope-intercept form, you must first solve the equation of the line for \(y\).
This will give you a clearer picture of the slope and y-intercept.
In this exercise, the original line equation \(2x + 5y + 8 = 0\) was converted to slope-intercept form by isolating \(y\).
We rearranged the equation to get \(y = -\frac{2}{5}x - \frac{8}{5}\), showing us that the slope \(m\) is \(-\frac{2}{5}\). By rewriting equations in this way, you can quickly identify and compare slopes and y-intercepts between different lines.

Point-Slope Form

The point-slope form of a line's equation is another useful way to describe a line when you know a point on the line and its slope.
The formula is \(y - y_1 = m(x - x_1)\), where:

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

This form is quite handy when you are given some point and need to write an equation for the line.
In our task, we used the point-slope form with the point \((-1, -2)\) and the slope \(\frac{5}{2}\) (calculated as the negative reciprocal of the original line's slope).
Substituting these into the formula gave us \(y + 2 = \frac{5}{2}(x + 1)\).
This form is great because it allows you to plug in any \(x\) value to instantly find \(y\).

Perpendicular Lines

Perpendicular lines intersect at a right angle (90 degrees).
One key property of perpendicular lines in a coordinate plane is the relationship between their slopes.
If two lines are perpendicular, the product of their slopes will be \(-1\).
The slope of the line perpendicular to another is found by taking the negative reciprocal of the other line’s slope. For instance, in this problem, we found the line with slope \(-\frac{2}{5}\), so the slope of the perpendicular line was \(\frac{5}{2}\).
This negative reciprocal relationship is fundamental when analyzing and working with perpendicular lines.
This simple yet powerful concept allows you to determine the necessary slope adjustments for creating perpendicular intersections.

Slope Calculation

Slope is a measure of how steep a line is, compared to the horizontal.
It represents the rate of change of \(y\) with respect to \(x\).
You can calculate slope \(m\) using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line.In our exercise, we already had the slope of the given line by arranging it into slope-intercept form.
In real-world problems or additional tasks, you might have to calculate it directly using the slope formula above.
Understanding how to manipulate equations to find slopes ensures clarity in graph interpretation and geometric problem-solving.

• Positive slopes mean the line rises as \(x\) increases.
• Negative slopes mean the line falls as \(x\) increases.

Learning these nuances will aid significantly in tackling complex geometry problems.