Problem 77 - Algebra for College Students

Question

Find the equation of line \(l\). Write the answer in standard form with integral coefficient with a positive coefficient for \(x .\) See Example 8. Line \(l\) goes through \((3,2)\) and is perpendicular to \(3 x-12 y=1\).

Step-by-Step Solution

Answer

The equation of line l in standard form is \(4x + y = 14\).

Step 1: Identify the slope of the given line

Rewrite the given equation in slope-intercept form. The equation is: \[3x - 12y = 1\]Solve for y to get it in the form \(y = mx + b\). First, isolate y: \[12y = 3x - 1\] Divide by 12: \[y = \frac{3}{12}x - \frac{1}{12}\]Simplify the fraction:\[y = \frac{1}{4}x - \frac{1}{12} \] The slope (m) of the original line is \(\frac{1}{4}\).

Step 2: Determine the slope of the perpendicular line

The slope of a line perpendicular to another line is the negative reciprocal of the original line’s slope. Therefore, the slope of line \(l\) is: \[-4 = -\frac{1}{\frac{1}{4}}\]. So, the slope of line \(l\) is \(-4\)

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

Use the point \((3,2)\) and the slope \(-4\) in the point-slope form: \(y - y_1 = m(x - x_1)\) \(y - 2 = -4(x - 3)\) Distribute and simplify: \(y - 2 = -4x + 12\) Solve for y: \(y = -4x + 14 \)

Step 4: Convert the equation to standard form

Take the equation \(y = -4x + 14 \) and bring all terms to one side to get the standard form \(Ax + By = C\): \(4x + y = 14\). So the standard form is: \(4x + y = 14\). The coefficient of x is positive.

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is one of the most common ways to express a line's equation. It is written as:
\[ y = mx + b \]
Where:

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

The goal is to isolate y on one side of the equation. This results in an easy-to-read format that tells you quickly both the slope and the y-intercept of the line. In our step-by-step solution, we start with the equation of the given line:
\[ 3x - 12y = 1 \]
To convert this to slope-intercept form, we solve for y:
<1>First, isolate y:
\[ 12y = 3x - 1 \]
Then, divide every term by 12:
\[ y = \frac{3}{12}x - \frac{1}{12} \]
Finally, simplify the fractions:
\[ y = \frac{1}{4} x - \frac{1}{12} \]Now, we have the slope-intercept form, and we can see that the slope (m) is \(\frac{1}{4}\). This is crucial for finding the slope of a perpendicular line.

point-slope form

The point-slope form is another way to write the equation of a line. It is very useful when you know one point on the line and the slope. The form is written as:
\[ y - y_1 = m(x - x_1) \]
Where:

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

In our exercise, we have determined the slope of the perpendicular line to be -4, and we have a known point (3, 2). Plugging these into the point-slope form:<1>
\[ y - 2 = -4(x - 3) \]
This step highlights the simplicity of using point-slope form when exact coordinates and slope are known. From here, we can distribute 1-4 on the right side:
\[ y - 2 = -4x + 12 \]
Finally, solve for y to convert into slope-intercept form:
\[ y = -4x + 14 \]
With the equation now in slope-intercept form, we can find the line's slope and y-intercept and use this information for additional transformations.

standard form

The standard form of a linear equation is another method of writing the line equation. This form is given by:
\[ Ax + By = C \]
Where:

• A, B, and C are integers
• A should be a positive integer

Let's convert our line equation, currently in slope-intercept form \( y = -4x + 14 \) to standard form:
1. Move all terms to one side to arrange in the format Ax + By = C:

• 
  \[ y = -4x + 14 \]
• 
  Add 4x to both sides:

• \[ 4x + y = 14 \]
  

We now have the equation in standard form where:

• A = 4 (positive integer as required)
• B = 1
• C = 14

This standard form representation is particularly useful in many mathematical applications, such as quickly identifying the values of the coefficients, simplifying evaluations, or setting up systems of equations.