Problem 35

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through $(1,3)$ and parallel to the horizontal line passing through $(3,-1)$

Step-by-Step Solution

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Answer

The equation is $y = 3$.

1Step 1: Identify the slope

A horizontal line has a slope of 0. Since the line must be parallel to the horizontal line through $(3, -1)$, it will also have a slope of 0.

2Step 2: Use point-slope form

Use the point-slope form of a line's equation, which is $y - y_1 = m(x - x_1)$. Substitute $m = 0$ (the slope), $x_1 = 1$, and $y_1 = 3$. This gives $y - 3 = 0(x - 1)$. After simplifying, we have $y = 3$.

3Step 3: Convert to standard form

The standard form of a line's equation is $Ax + By = C$. Since $y = 3$, this can be rewritten as $0x + 1y = 3$, which simplifies to $y = 3$. Thus, the standard form of the equation is already $y = 3$.

Key Concepts

Point-slope formStandard form of a lineSlope of a line

Point-slope form

The point-slope form of a linear equation is a tool used to find the equation of a line when you know one point on the line and the slope. It is expressed as:

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

Here, $ (x_1, y_1) $ represents a known point on the line, and $ m $ is the slope. This form is particularly handy because it directly connects the slope to a specific point, making it easy to plug in numbers and solve.
The process involves:

Identifying the slope, which in this case is 0 for a horizontal line.
Choosing the given point, which is $(1, 3)$.

Substitute these values into the point-slope formula. It transforms the equation into a practical form that can further be simplified or converted into other forms, such as the standard form.

Standard form of a line

The standard form of a line is one of the common ways to express linear equations. Its formula is:

$ Ax + By = C $

In this equation:

$ A, B, \text{ and } C $ are integers.
$ A \geq 0 $.

The main advantage of the standard form is its versatility in representing vertical and horizontal lines clearly. For horizontal lines, like the line $ y = 3 $, the standard form is simply $0x + 1y = 3$.
Converting from another form, such as slope-intercept

Start with the simplified equation $ y = 3 $.
Recognize this already follows the $ Ax + By = C $ pattern with $ B = 1 $ and $ C = 3 $.

This flexibility in conversion makes the standard form a reliable choice in various problem scenarios.

Slope of a line

The slope of a line is a measure of its steepness and direction. It's calculated as the ratio of the change in the vertical coordinates to the change in the horizontal coordinates between two distinct points:

$ m = \frac{y_2 - y_1}{x_2 - x_1} $

In this exercise, however, the line is parallel to a horizontal line.
For any horizontal line, the slope (

$ m $) is 0
because the vertical change is zero.

Whenever a line is parallel to a horizontal line, its slope remains 0, indicating no incline. This property simplifies many calculations, as it means the line is parallel to the x-axis, leading directly to equations like $ y = C $ without any x-component, indicating constant output regardless of x. This simplicity helps maintain focus on problem-solving without complex computations.

Problem 35

Problem 36

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