Problem 40 - Calculus with Concepts in Calculus

Question

Find an equation of the line that is parallel to the given line \(l\) and passes through the given point \(P\). \(l: 3 y+2 x=5 ; P=(-1,-3)\)

Step-by-Step Solution

Answer

The equation is \( y = -\frac{2}{3}x - \frac{11}{3} \).

Step 1: Identify the Slope of the Given Line

First, we need to rewrite the equation of line \( l : 3y + 2x = 5 \) in slope-intercept form \( y = mx + b \) to find its slope. Start by solving for \( y \):\[ 3y = -2x + 5 \]\[ y = -\frac{2}{3}x + \frac{5}{3} \]The slope \( m \) of the line \( l \) is \( -\frac{2}{3} \). Because parallel lines have the same slope, the new line we are going to find will have this same slope.

Step 2: Use the Point-Slope Formula

Now that we have the slope of the desired line, \( m = -\frac{2}{3} \), and a point it goes through, \( P(-1, -3) \), we can use the point-slope formula, \( y - y_1 = m(x - x_1) \), to find the equation. Substitute \( m = -\frac{2}{3} \), \( x_1 = -1 \), and \( y_1 = -3 \):\[ y - (-3) = -\frac{2}{3}(x - (-1)) \]This simplifies to:\[ y + 3 = -\frac{2}{3}(x + 1) \].

Step 3: Simplify the Equation

Now, let's simplify the equation to its slope-intercept form, \( y = mx + b \). Distribute \( -\frac{2}{3} \) through \( (x + 1) \):\[ y + 3 = -\frac{2}{3}x - \frac{2}{3} \]To isolate \( y \), subtract 3 from both sides:\[ y = -\frac{2}{3}x - \frac{2}{3} - 3 \]Convert \( 3 \) to a fraction with a common denominator of 3, \( 3 = \frac{9}{3} \), to combine:\[ y = -\frac{2}{3}x - \frac{2}{3} - \frac{9}{3} \]Combine like terms:\[ y = -\frac{2}{3}x - \frac{11}{3} \]The equation of the line parallel to \( l \) through point \( P \) is \( y = -\frac{2}{3}x - \frac{11}{3} \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line's equation makes it very easy for us to understand two important features: the slope and the y-intercept. This form is written as \( y = mx + b \), where:

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

The slope \( m \) tells us how steep the line is, with a positive value indicating the line rises as it moves from left to right, and a negative value showing it falls.
Slope-intercept form is very handy, especially for graphing lines. All you need is the slope and the y-intercept to sketch it, which makes understanding and visualizing linear equations a breeze!

Point-Slope Formula

The point-slope formula is a useful tool when you have a line's slope and a point on the line. The formula is given by \( y - y_1 = m(x - x_1) \), where:

• \( m \) is the slope of the line
• \((x_1, y_1)\) are the coordinates of a given point on the line

This formula is particularly helpful for finding the equation of a line when you're not starting with it in slope-intercept form but have a point that the line passes through.
Using this approach, you can quickly generate an equation from known values, simplifying the process of working with points and slopes. It bridges the gap between facts you know and the equation you want, making calculations smoother.

Parallel Lines

Parallel lines are lines in a plane that never meet. They stay the same distance apart over their entire length. An important mathematical property of parallel lines is that they share the same slope.

• If a line has the equation \( y = mx + b \), any line parallel to it will have the slope \( m \).
• Changing the y-intercept \( b \) results in a parallel line, maintaining the direction but varying where it crosses the y-axis.

In problems involving line equations, identifying parallel lines means ensuring the slopes are equal.
This concept is crucial in geometry and graphing, as it helps understand the spatial relations between different lines without needing to see them drawn.