Problem 32 - Contemporary Precalculus

Question

Find the equation of the line with slope \(m\) that passes through the given point. $$m=0 ;(-4,-5)$$

Step-by-Step Solution

Answer

Answer: The equation of the line is y = -5.

Step 1: Identify the given point and slope

The given point is \((-4,-5)\) and the given slope is \(m=0\).

Step 2: Plug the point and slope into the point-slope equation

We will now plug the given point \((-4,-5)\) and the slope \(m=0\) into the point-slope form equation, \(y - y_1 = m(x - x_1)\). This will give us: $$y - (-5) = 0(x - (-4))$$

Step 3: Simplify the equation

Now, we will simplify the equation from step 2: $$y + 5 = 0(x + 4)$$ Since multiplying any value by 0 results in 0, the equation simplifies to: $$y + 5 = 0$$

Step 4: Solve for y

Finally, we will isolate the variable \(y\) by subtracting 5 from both sides of the equation: $$y = -5$$ This is the equation of the line with slope \(m=0\) that passes through the point \((-4, -5)\).

Key Concepts

Point-Slope Form

One of the most useful forms for writing the equation of a line is the point-slope form. This form is especially helpful when you know a point on the line and the slope. The formula is:

• \(y - y_1 = m(x - x_1)\)

Here, \((x_1, y_1)\) is a known point, and \(m\) is the slope of the line.
To use this formula, you'll simply substitute the values of \(x_1\), \(y_1\), and \(m\) into the equation. In the given problem, the point is \((-4, -5)\) and the slope \(m\) is 0.
Substituting these values into the equation, we first find:

• \(y - (-5) = 0(x - (-4))\)

This approach ensures we have integrated the point's coordinates and slope into developing the equation of the line.

Slope

The slope is a critical concept in understanding linear equations and graphing lines. It represents the steepness and direction of a line.
Slope is commonly denoted by \(m\) and is calculated as the ratio of the change in \(y\) (vertical change) to the change in \(x\) (horizontal change) between two points on a line. The formula is:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

In our specific exercise, the slope \(m\) is already provided as 0, indicating a horizontal line.
A slope of 0 implies that no matter how far along the \(x\) axis we move, the \(y\) value doesn't change; hence the line is flat. Knowing the slope allows us not only to draw the line correctly but also to easily formulate its equation using the point-slope form.

Linear Equations

Linear equations describe straight lines and are pivotal in algebra. They can be represented in various forms, such as point-slope, slope-intercept, or standard form.
In this exercise, we transformed the point-slope form into a simplified equation. After inserting our point \((-4, -5)\) and slope \(0\) into the equation, we simplified it to \(y = -5\).
The final equation \(y = -5\) exemplifies a horizontal line where \(y\) is consistently -5, showing that no matter the \(x\) value, \(y\) remains unchanged.

• Slope-Intercept Form: \(y = mx + b\)
• Standard Form: \(Ax + By = C\)

These forms make it possible to analyze and graph linear relationships effectively, serving as fundamental tools for solving real-world problems.