Problem 35 - Calculus Single Variable

Question

Write the slope-intercept equation of the line that passes through the given point and that is perpendicular to the given line. $$ (3,4), x / 3+y / 2=1 $$

Step-by-Step Solution

Answer

The slope-intercept equation is \( y = \frac{3}{2}x - \frac{1}{2} \).

Step 1: Convert the given line to slope-intercept form

The given line equation is \( \frac{x}{3} + \frac{y}{2} = 1 \). To convert it to slope-intercept form (\( y = mx + b \)), solve for \( y \).First, subtract \( \frac{x}{3} \) from both sides:\[ \frac{y}{2} = -\frac{x}{3} + 1 \]Next, multiply every term by 2 to clear the fraction:\[ y = -\frac{2}{3}x + 2 \]So the line's slope is \( m = -\frac{2}{3} \).

Step 2: Determine the slope of the perpendicular line

Lines that are perpendicular have slopes that are negative reciprocals of each other. The original line's slope is \( -\frac{2}{3} \), so the perpendicular slope is the negative reciprocal:\[ m = \frac{3}{2} \].

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

We use the point-slope form equation \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (3, 4) \) and \( m = \frac{3}{2} \):\[ y - 4 = \frac{3}{2}(x - 3) \].

Step 4: Convert to slope-intercept form

Distribute \( \frac{3}{2} \) across \( (x - 3) \):\[ y - 4 = \frac{3}{2}x - \frac{9}{2} \]Add 4 to both sides to solve for \( y \):\[ y = \frac{3}{2}x - \frac{9}{2} + 4 \].Convert 4 to a fraction with a denominator of 2: \( 4 = \frac{8}{2} \), giving us:\[ y = \frac{3}{2}x - \frac{1}{2} \].

Key Concepts

Slope-Intercept Form

When we talk about the slope-intercept form in algebra, we refer to a way of expressing the equation of a straight line. This is given in the formula \( y = mx + b \). Here, \( m \) represents the slope of the line, which describes its steepness or tilt. Meanwhile, \( b \) indicates the y-intercept, which is the point where the line crosses the y-axis.
Converting an equation like \( \frac{x}{3} + \frac{y}{2} = 1 \) into slope-intercept form involves isolating \( y \) on one side of the equation. This process not only makes it easier to plot the line on a graph, but also allows us to clearly see its slope and y-intercept.

• The slope of the line dictates whether the line rises or falls as it moves from left to right. Positive slopes rise and negative slopes fall.
• The y-intercept provides a starting point on the y-axis, anchoring the line in place.

Understanding the slope-intercept form is essential for graphing linear equations and predicting how changes in slope or intercept affect the line's position or orientation.

Point-Slope Form

The point-slope form is another powerful tool in understanding and working with linear equations. This form is best used when you know a specific point on the line and the line's slope. It is given by the equation \( y - y_1 = m(x - x_1) \). In this formula, \((x_1, y_1)\) represents a known point on the line, and \(m\) is the slope.
Point-slope form is incredibly useful when we're given a particular point that the line passes through, especially during problems involving finding equations of lines parallel or perpendicular to other lines.

• By substituting the known values into the equation, we gain a direct expression of the line in question.
• This form is straightforward for transitioning into the slope-intercept form, where the simplified form of the line can be easily graphically represented or applied in further calculations.

This form efficiently connects a known piece of information (a point) with some change factor (the slope), giving a clear definition of a line in contextual scenarios.

Negative Reciprocals

Negative reciprocals are fundamental when dealing with perpendicular lines. If two lines are perpendicular, their slopes are negative reciprocals of each other. This means that the product of their slopes is \(-1\).
For example, if a line has a slope \( m = -\frac{2}{3} \), the perpendicular line will have a slope of \( \frac{3}{2} \). We achieve this by flipping the fraction (taking the reciprocal) and then changing its sign (making it negative or positive, as needed).

• This concept helps us solve problems where we need to find a line that is perpendicular to a given line.
• Knowing the slope of the original line immediately allows us to calculate the slope of its perpendicular counterpart.

Mastering the concept of negative reciprocals helps in quickly identifying relationships between lines on a graph and is pivotal in problems involving geometric and algebraic concepts of perpendicularity.