Problem 82 - 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 \(l\) has slope 5 and goes through \(\left(0, \frac{1}{2}\right)\).

Step-by-Step Solution

Answer

The standard form is \(10x - 2y = -1\).

Step 1: Find the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Given the slope \(m = 5\) and the line passes through \(\left(0, \frac{1}{2}\right)\), substitute \(m\) and \( (x, y) = (0, \frac{1}{2})\) into the equation to solve for \(b\): \( \frac{1}{2} = 5 \cdot 0 + b \).

Step 2: Solve for the Y-intercept

From the previous step, we have \( \frac{1}{2} = b \). Therefore, the y-intercept \(b\) is \( \frac{1}{2} \). So the equation in slope-intercept form is: \( y = 5x + \frac{1}{2} \).

Step 3: Convert to Standard Form

To write the equation in standard form with integral coefficients, start with the equation \( y = 5x + \frac{1}{2} \). Multiply every term by 2 to clear the fraction: \( 2y = 10x + 1 \). Then rearrange it into standard form \( Ax + By = C \): \( -10x + 2y = 1 \). Or, optionally, multiply by -1 to get positive coefficients for \( Ax \): \( 10x - 2y = -1 \).

Key Concepts

slope-intercept form

The slope-intercept form of a line is incredibly useful for quickly identifying the slope and y-intercept of a line. It is written as:
y = mx + b

where:**m** represents the slope, which tells us how steep the line is, while **b** is the y-intercept, the point where the line crosses the y-axis.
In our exercise, we are given the slope, **m = 5**, and a point the line passes through, **(0, \(\frac{1}{2}\))**. By substituting these values into the slope-intercept form, we can find the y-intercept.
Firstly, we substitute into the slope-intercept form: \(\frac{1}{2} = 5 \times 0 + b\).
Solving for **b**, we find that **b = \(\frac{1}{2}\)**. Hence, the slope-intercept form of our line is:

\( y = 5x + \frac{1}{2}\)

This form is straightforward and shows both the slope and y-intercept, making it easy to graph the line or understand its general direction and position.

standard form

The standard form of a line is another way to represent a linear equation. It is written as:
Ax + By = C

where **A**, **B**, and **C** are integers and **A** should ideally be non-negative.

Standard form is useful in certain applications, particularly in systems of linear equations. To convert from slope-intercept form to standard form, we rearrange the equation and clear any fractions.

Starting from our slope-intercept form: \( y = 5x + \frac{1}{2}\),
we multiply every term by 2 to eliminate the fraction:

\( 2y = 10x + 1 \)

Then, we rearrange it to get all variable terms on one side:

\( -10x + 2y = 1 \)

Since the coefficient of **x** is negative, we can multiply the entire equation by -1 to make it positive:

\( 10x - 2y = -1 \)
Thus, in standard form, the equation of our line is:

\( 10x - 2y = -1 \)

integral coefficients

To ensure a linear equation is in the preferred standard form, **Ax + By = C**, with integral coefficients, follow these steps:

• Clear any fractions by multiplying through by the least common multiple (LCM) of the denominators
• Ensure **A**, **B**, and **C** are integers (whole numbers)
• Adjust the equation so that **A** is positive, if necessary

In our exercise, we started with: \( y = 5x + \frac{1}{2}\).
Multiplying all terms by 2 cleared the fraction: \( 2y = 10x + 1 \).
Rearranging for standard form gave: \( -10x + 2y = 1 \). Multiplying through by -1 resulted in: \( 10x - 2y = -1 \), with integral coefficients for **A**, **B**, and **C**.
Finally, these steps ensure the equation conforms to the standard form requirements with all terms as integers, and **A** being positive, simplifying it for practical use in further mathematical applications.