Problem 92 - Algebra for College Students

Question

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) is perpendicular to \(2 y+5-3 x=0\) and goes through \((2,7)\).

Step-by-Step Solution

Answer

The equation of line \(l\) is \(2x + 3y = 25\).

Step 1: Rewrite the given line in slope-intercept form

First, rewrite the given line equation, \(2y + 5 - 3x = 0\), in slope-intercept form, which is \(y = mx + b\)Subtract 5 from both sides:\(2y = 3x - 5\)Then, divide by 2:\(y = \frac{3}{2}x - \frac{5}{2}\)The slope (\(m\)) of the given line is \( \frac{3}{2} \).

Step 2: Find the perpendicular slope

The slopes of perpendicular lines are negative reciprocals of each other. Thus, if the slope of the given line is \( \frac{3}{2} \), the slope of the line perpendicular to it will be \( -\frac{2}{3} \).

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

Given the slope \( m = -\frac{2}{3} \) and the point (2, 7), use the point-slope form: \( y - y_1 = m(x - x_1) \)Substituting in the point and the slope:\( y - 7 = -\frac{2}{3}(x - 2) \).

Step 4: Simplify and write in slope-intercept form

Distribute the slope on the right-hand side:\( y - 7 = -\frac{2}{3}x + \frac{4}{3} \)Add 7 to both sides to isolate \(y\):\( y = -\frac{2}{3}x + \frac{4}{3} + 7 \)Converting 7 to a fraction gives:\( y = -\frac{2}{3}x + \frac{4}{3} + \frac{21}{3} \)Combining like terms, we get:\( y = -\frac{2}{3}x + \frac{25}{3} \).

Step 5: Convert to standard form with integral coefficients

Multiply through by 3 to clear the denominators:\( 3y = -2x + 25 \)Rearrange into standard form \(Ax + By = C\):\( 2x + 3y = 25 \).

Key Concepts

slope-intercept form

The slope-intercept form of a line is a way to express the equation of a line so that it’s easy to see both the slope and the y-intercept. The formula is written as:

\( y = mx + b \)

• m is the slope of the line. The slope measures how steep the line is and the direction it is going. It’s calculated as the 'rise over run,' or the change in y divided by the change in x.


• b is the y-intercept, the point where the line crosses the y-axis.

Converting an equation like \(2y + 5 - 3x = 0\) into slope-intercept form involves solving for \(y\) in terms of \(x\). For example:

1. Subtract 5 from both sides: \( 2y = 3x - 5 \)

2. Divide by 2: \( y = \frac{3}{2}x - \frac{5}{2} \)

point-slope form

The point-slope form is another way to write the equation of a line when you know a point on the line and its slope. The formula is:

\( y - y_1 = m(x - x_1) \)

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

For example, if we know the slope is \( -\frac{2}{3} \) and it passes through the point (2, 7), we substitute these values into the formula:

\( y - 7 = -\frac{2}{3}(x - 2) \).
Then, we simplify it for convenience.
For instance, distributing the slope and adjusting for y, we get the equation in slope-intercept form.

This will transform our example to: \(y = -\frac{2}{3}x + \frac{25}{3} \) after necessary steps.

standard form

The standard form of the equation of a line is written as:

\( Ax + By = C \)

• A , B , and C are integers.


• It's a very common form, especially when working with linear systems of equations.

To convert a line equation into standard form, clear any fractions and rearrange the terms properly. For instance, from the example above: \( y = -\frac{2}{3}x + \frac{25}{3} \), we multiply through by 3 to get: \( 3y = -2x + 25 \).

Finally, rearrange it:

\( 2x + 3y = 25 \).
This represents the same line in standard form.