Problem 81 Find the equation of line l in e... [FREE SOLUTION]

Chapter 3: Problem 81

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) has slope \(\frac{1}{2}\) and goes through \((0,5)\).

Short Answer

Expert verified

The equation in standard form is \( -x + 2y = 10 \).

Step by step solution

Understand the Slope-Intercept Form

The slope-intercept form of a line's equation is given by: \( y = mx + b \)where \(m\) is the slope and \(b\) is the y-intercept.

Identify the Slope and Y-Intercept

From the problem, the slope \( m \) is \( \frac{1}{2} \) and the line goes through the point \( (0, 5) \). This point is the y-intercept \( b \), so \( b = 5 \).

Write the Equation in Slope-Intercept Form

Substitute the slope and y-intercept into the slope-intercept form: \( y = \frac{1}{2}x + 5 \)

Convert to Standard Form

To convert \( y = \frac{1}{2}x + 5 \) to standard form, multiply every term by 2 to eliminate the fraction: \( 2y = x + 10 \)Then, rearrange to get the standard form: \( -x + 2y = 10 \)

Ensure Integral Coefficients

Make sure all coefficients are integers. Here, the coefficients are already integers: \( -x + 2y = 10 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form

The slope-intercept form is a popular method for writing the equation of a line. This form is written as \( y = mx + b \), where:

\( m \) represents the slope of the line, which is the rate at which the line rises or falls.
\( b \) represents the y-intercept, which is the point where the line crosses the y-axis.

To understand these concepts better, let's break them down:

**Slope**, \( m \), is calculated as the ratio of the vertical change to the horizontal change between two points on the line. For example, a slope of \(\frac{1}{2}\) means that for every 2 units moved horizontally, the line moves 1 unit vertically.
**Y-intercept**, \( b \), is simply the value of \( y \) when \( x \) is 0. This is where the line meets the y-axis.

In our exercise, the slope \( m \) is \(\frac{1}{2}\), and since the line crosses the y-axis at point \( (0, 5) \), \( b \) equals 5. Therefore, plugging in these values, the slope-intercept form of the equation is \( y = \frac{1}{2} x + 5 \).

standard form

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

\( A \), \( B \), and \( C \) are integers.

Steps to convert from slope-intercept to standard form:

Identify the current form of the equation. Starting with our slope-intercept form: \( y = \frac{1}{2}x + 5 \).
Eliminate any fractions by multiplying through by the denominator. In our case, multiply every term by 2: \( 2y = x + 10 \).
Arrange the equation in its standard form: \( Ax + By = C \). Here it becomes: \( -x + 2y = 10 \).

Make sure: \( A \), \( B \), and \( C \) are integers (no decimals or fractions) and usually, \( A \) should be a positive integer. If we want \( A \) to be positive, we can multiply through by -1, resulting in \( x - 2y = -10 \).
In our exercise, \( -x + 2y = 10 \) is already in standard form, and the coefficients are integers.

integral coefficients

Integral coefficients mean that the numbers for \( A \), \( B \), and \( C \) should be whole numbers. This ensures the equation is simplified without fractions or decimals. Here are some steps to ensure integral coefficients:

**Eliminate Fractions**: Multiply every term by the least common multiple of the denominators.
**Simplify Integers**: Rearrange the equation to ensure all coefficients are integers and often ensure \( A \) is positive if needed.
**Check Your Work**: Verify that there are no fractions or negative signs where unnecessary.

For example, starting with the slope-intercept form from above, \( y = \frac{1}{2}x + 5 \). By multiplying every term by 2, we eliminate the fraction: \( 2y = x + 10 \). Finally, rearranging gives us: \( -x + 2y = 10 \). The coefficients \( -1 \), \( 2 \), and \( 10 \) are all integers, thus satisfying the requirement for integral coefficients.

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91影视

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) has slope \(\frac{1}{2}\) and goes through \((0,5)\).

Short Answer

Step by step solution

Understand the Slope-Intercept Form

Identify the Slope and Y-Intercept

Write the Equation in Slope-Intercept Form

Convert to Standard Form

Ensure Integral Coefficients

Key Concepts

slope-intercept form

standard form

integral coefficients

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