Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 45-48, find the coordinates of the point of intersection. Then write an equation for the line through that point perpendicular to the line given first. \(5 x-2 y=5\) \(2 x+3 y=6\)

Step-by-Step Solution

Answer

Intersection at \(\left( \frac{27}{19}, \frac{20}{19} \right)\); perpendicular line: \(y = -\frac{2}{5}x + \frac{374}{95}\)."

Step 1: Rewrite the Equations in Slope-Intercept Form

First, we want to express both equations in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For the first equation, \(5x - 2y = 5\):\[2y = 5x - 5\]\[y = \frac{5}{2}x - \frac{5}{2}\]For the second equation, \(2x + 3y = 6\): \[3y = -2x + 6\] \[y = -\frac{2}{3}x + 2\]

Step 2: Find the Intersection Point

To find the point of intersection of the two lines, set the equations for \(y\) equal to each other:\[\frac{5}{2}x - \frac{5}{2} = -\frac{2}{3}x + 2\]Clear the fractions by multiplying the entire equation by 6:\[15x - 15 = -4x + 12\]Combine like terms:\[19x = 27\]Solve for \(x\):\[x = \frac{27}{19}\]Substitute back into one of the equations, let's use \(y = \frac{5}{2}x - \frac{5}{2}\):\[y = \frac{5}{2}(\frac{27}{19}) - \frac{5}{2}\]\[y = \frac{135}{38} - \frac{95}{38}\]\[y = \frac{40}{38} = \frac{20}{19}\]So, the point of intersection is \(\left( \frac{27}{19}, \frac{20}{19} \right)\).

Step 3: Determine the Slope of the Perpendicular Line

The slope of the first line is \(\frac{5}{2}\). A perpendicular line has a slope that is the negative reciprocal.So, the slope \(m\) of the perpendicular line is \[m = -\frac{2}{5}\].

Step 4: Write the Equation of the Perpendicular Line

To find the equation of the perpendicular line, use point-slope form: \[y - y_1 = m(x - x_1)\]Using the point of intersection \(\left( \frac{27}{19}, \frac{20}{19} \right)\) and \(m = -\frac{2}{5}\): \[y - \frac{20}{19} = -\frac{2}{5}(x - \frac{27}{19})\]Distribute and simplify:\[y = -\frac{2}{5}x + \frac{54}{95} + \frac{20}{19}\]Finding a common denominator and combining: \[y = -\frac{2}{5}x + \frac{374}{95}\]This is the equation of the line perpendicular to the first through the intersection point.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is one of the most common ways to write the equation of a straight line. This form is especially useful because it clearly shows two key pieces of information about a line:

• The slope of the line, represented by the letter \( m \).
• The y-intercept, represented by the letter \( b \).

In the equation \( y = mx + b \), the slope \( m \) describes how steep the line is, while the y-intercept \( b \) is where the line crosses the y-axis.

To convert an equation into slope-intercept form, solve the equation for \( y \). This means rearranging the equation so \( y \) is on one side and the rest of the terms are on the other side. For instance, the equation \( 5x - 2y = 5 \) can be rewritten as \( y = \frac{5}{2}x - \frac{5}{2} \), showing a slope of \( \frac{5}{2} \) and a y-intercept of \( -\frac{5}{2} \). Similarly, the equation \( 2x + 3y = 6 \) becomes \( y = -\frac{2}{3}x + 2 \), with a slope of \( -\frac{2}{3} \) and a y-intercept of \( 2 \).
Understanding slope-intercept form is essential as it lays the groundwork for finding key characteristics of lines quickly, especially when analyzing or graphing them.

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle, or 90 degrees. In the context of slopes, two lines are perpendicular if the product of their slopes is \(-1\). This means that if one line has a slope of \( m \), the slope of a line perpendicular to it will be \(-\frac{1}{m}\).

Let's take an example from the given problem: the line with the slope \( \frac{5}{2} \). A line perpendicular to this would have a slope that's the negative reciprocal: \(-\frac{2}{5}\). This relationship makes it easy to find the slopes of perpendicular lines directly from each other.

Understanding and identifying perpendicular lines is crucial for geometry and algebra. It helps in solving problems related to angles and intersections. When given a line and asked to find a perpendicular one, remember that flipping and negating the slope is your key!
Always ensure to verify your work, as correct calculation of slopes is fundamental in finding accurate perpendicular line equations.

Point-Slope Form

The point-slope form is another important equation format that is very handy when you know a point on the line and its slope. The general formula for point-slope form is \(y - y_1 = m(x - x_1)\), where:

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

This form is particularly useful for finding the equation of a line when you're given any point on the line and the slope. It is also very helpful in deriving the equation of a line when dealing with perpendicular lines.

Consider our problem, where the intersection point is \(\left( \frac{27}{19}, \frac{20}{19} \right)\) and the required slope for a perpendicular line is \(-\frac{2}{5}\). Using these in the point-slope form yields \(y - \frac{20}{19} = -\frac{2}{5}(x - \frac{27}{19})\). This makes it easy to create an equation direct from both known values without having to rearrange or substitute too much.

Understanding how to effectively use the point-slope form can save time and simplify many tasks involving linear equations. It's a powerful tool especially when working through problems involving line intersections or perpendiculars.