Problem 21 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given point and has the given slope. $$ (-6,2), m=-5 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y - 2 = -5(x + 6)\)

Step 1: Identify the Given Values

From the exercise we know that the given point is (-6,2) and the slope, denoted as \(m\), is -5.

Step 2: Substitute the Given Values into the Point-Slope Form

The point-slope form of a line is given by the formula \(y - y_1 = m(x - x_1)\). Here, \(x_1\) is -6, \(y_1\) is 2 and \(m\) is -5. Substituting these values into the formula, we get: \(y - 2 = -5(x + 6)\)

Step 3: Simplify the Equation

The equation \(y - 2 = -5(x + 6)\) is already in the point-slope form. Thus, no further simplification is necessary and this is the equation of the line.

Key Concepts

Equation of a Line

The equation of a line is a mathematical representation that describes all the points lying on that line. It establishes a relationship between the coordinates on the x-axis (horizontal) and y-axis (vertical).

There are several forms used to express a line's equation, and each form is suitable for different scenarios. The most common forms include:

• **Slope-Intercept Form:** Written as \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
• **Point-Slope Form:** Useful when you know a point on the line and its slope. It's expressed as \(y - y_1 = m(x - x_1)\).
• **Standard Form:** Presented as \(Ax + By = C\), where \(A, B,\) and \(C\) are integers.

For the given exercise, the point-slope form is the most fitting. This form is particularly helpful for writing the line's equation when you have a specific point and the line's slope.

Slope

The slope is a crucial concept when dealing with linear equations. It measures the steepness or incline of a line, essentially telling us how much the line goes up or down for each step it goes across.

The formula to calculate slope is \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1, y_1\) and \(x_2, y_2\) are coordinates of two points on the line. In the point-slope form \(y - y_1 = m(x - x_1)\), \(m\) represents the slope.

Some key features of slope include:

• A positive slope means the line rises from left to right.
• A negative slope indicates the line falls from left to right.
• A zero slope defines a horizontal line.
• An undefined slope describes a vertical line.

In this exercise, the slope is given as \(-5\), suggesting the line decreases sharply as it moves from left to right.

Coordinates of a Point

Coordinates define the specific location of a point on a graph. They are written as pairs, typically represented as \(x, y\). The x-coordinate tells us how far along the x-axis the point is, while the y-coordinate shows its position along the y-axis.

For example, in the pair \((-6, 2)\), the x-coordinate is \(-6\) and the y-coordinate is \(+2\). Here's what these numbers mean:

• The point is 6 units to the left of the origin on the x-axis.
• The point is 2 units above the origin on the y-axis.

Understanding coordinates is very important when working with the point-slope form of a line. They let you plug the values directly into the formula, showing precisely where the line should pass through in the graph.