Problem 18 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(--2,\) passing through \((0,-3)\)

Step-by-Step Solution

Answer

The point-slope form of the line is \(y + 3 = -2x\). The slope-intercept form of the line is \(y = -2x -3\).

Step 1: Write the point-slope form

The point-slope formula is given by: \(y - y1 = m(x - x1)\), where \(m\) is the slope, \(x1\) and \(y1\) are coordinates of the point. Given slope \(m\) is \(-2\) and the point is \((0, -3)\), so \(x1 = 0\) and \(y1 = -3\). Plugging these values into the formula we get: \(y - (-3) = -2(x - 0)\).

Step 2: Simplify the point-slope form equation

We simplify the equation \(y - (-3) = -2(x - 0)\) to: \(y + 3 = -2x\). This is the point-slope form of the equation.

Step 3: Transform to slope-intercept form

The slope intercept formula is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. We rearrange the simplified form \(y + 3 = -2x\) to this format which gives us: \(y = -2x -3\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the simplest and most common ways to represent the equation of a line. It is written as \(y = mx + b\). Here, \(m\) represents the slope of the line, while \(b\) represents the y-intercept, or the point where the line crosses the y-axis.
Understanding the slope-intercept form is incredibly useful because it provides a direct way to graph a line. You simply need to identify the slope \(m\) and the intercept \(b\) and you can draw the line on a graph.

• Start plotting from point \((0, b)\) on the y-axis.
• Use the slope \(m\) to determine how to rise and run from the intercept to find another point on the line.

With this form, you can easily interpret the behavior of the line just by looking at \(m\) and \(b\). This makes it a favorite among many students and professionals alike.

Equation of a Line

The equation of a line can be expressed in various ways, depending on the information available. Two of the most common forms are the point-slope form and the slope-intercept form.
The point-slope form is useful when you know a point on the line and the slope. The formula is: \(y - y_1 = m(x - x_1)\). This form quickly translates a point and a slope into an equation.
The slope-intercept form \(y = mx + b\) is used to uncover more about the characteristics of a line, particularly how it intercepts the y-axis and its steepness.

• The point-slope form directly utilizes a single point and a slope.
• The slope-intercept form simplifies understanding the line's rise and run.
• Choosing the appropriate form depends on the given data and what you need to find.

Recognizing which form to use helps build confidence in graphing and solving linear equations.

Slope

The slope \(m\) of a line measures its steepness and direction. It tells you how much y increases for every unit increase in x and is calculated as the ratio of the vertical change to the horizontal change between two points on the line: \[m = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1}\]A positive slope means the line rises from left to right, a negative slope means it falls, zero slope means it's horizontal, and an undefined slope represents a vertical line.
When writing the equation of a line, the slope is integral to show how the line changes as you move along it. In both the point-slope form \(y - y_1 = m(x - x_1)\) and the slope-intercept form \(y = mx + b\), the slope \(m\) determines the angle of inclination of the line.

Coordinates

Coordinates are pairs of numerical values that define a point's location on a Cartesian plane. Written as \((x, y)\), these pairs show how far along the x-axis and y-axis a point is located.
When dealing with the equation of a line, having at least one coordinate is essential to uniquely specify it alongside the slope. For instance, in the point-slope formula \(y - y_1 = m(x - x_1)\), \((x_1, y_1)\) represents a known point through which the line passes.

• Coordinates help in positioning a line on a graph.
• A specific point is needed to translate slope data into a line format.
• Understanding both axes positions allows for precise graph plotting.

Coordinates become powerful when solving real-world problems, enabling the translation of abstract math into tangible diagrams.