Problem 8 - Introductory and Intermediate Algebra for College Students 4th

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises \(1-28 .\) Then use the point-slope form of the equation to write the slope-intercept form of the equation. Slope \(=-11,\) passing through \((0,-3)\)

Step-by-Step Solution

Answer

The slope-intercept form of the equation is \(y = -11x - 3\).

Step 1: Write the Point-Slope Form of the Equation

Given that the slope \(m = -11,\) and the point is \((0,-3),\) we substitute these values into the point-slope form of a line equation \(y - y1 = m(x - x1)\). Therefore, the equation becomes \(y - (-3) = -11(x - 0)\), which simplifies to \(y + 3 = -11x\).

Step 2: Convert equation from Point-Slope Form to Slope-Intercept Form

The slope-intercept form of the equation is \(y = mx + c,\) where c is the y-intercept. By moving terms around in the equation, we can change our equation into slope-intercept form. Start by subtracting 3 from both sides to get \(y = -11x - 3\).

Key Concepts

Slope-Intercept Form

In the world of linear equations, the slope-intercept form is a popular way to express the equation of a line. This form is as simple as it sounds: \[y = mx + c\]Here, \(m\) represents the slope of the line, and \(c\) is the y-intercept, the point where the line crosses the y-axis. It is an incredibly handy form because it allows anyone to easily identify the slope and y-intercept directly from the equation.For example, when a line has an equation like \(y = -11x - 3\), it tells us several things:

• The slope \(m\) is \(-11\), which indicates the line declines steeply. A greater negative slope implies a steeper decline.
• The y-intercept \(c\) is \(-3\), meaning the line meets the y-axis at the point \((0, -3)\).

To convert from point-slope form to slope-intercept form, you align the formula as follows: solve \(y - y_1 = m(x - x_1)\) for \(y\). This conversion provides a straightforward way to understand the line's behavior in the coordinate plane.

Equation of a Line

The equation of a line is like a mathematical sentence that describes a straight line in a coordinate plane. It tells us exactly how the line behaves and where it sits in two-dimensional space. There are many forms of line equations, but the most commonly discussed are the point-slope and slope-intercept forms.

• **Point-slope form**: This form is ideal when you know a point on the line and the slope. It looks like this: \[y - y_1 = m(x - x_1)\]where \(m\) is the slope, and \((x_1, y_1)\) is a point on the line.
• **Slope-intercept form**: Useful for quickly identifying the slope and y-intercept: \[y = mx + c\]where \(m\) is the slope and \(c\) is the y-intercept.

Understanding different forms of line equations helps in translating word problems into mathematical ideas, crafting graphs with precision, and solving for unknown variables with ease. Learning to swap between different forms, like converting from a point-slope to a slope-intercept, becomes crucial in solving various real-life and academic problems.

Linear Equations

Linear equations are the foundation of algebra, representing relationships with a consistent rate of change. These equations describe straight lines and are defined as first-degree equations, meaning the highest power of the variable is one.In its broadest sense, a linear equation can be simplified into the form: \[Ax + By + C = 0\]Here, \(A\), \(B\), and \(C\) are constants, and \(x\) and \(y\) are the variables. However, linear equations are often used in their more familiar forms, such as the slope-intercept form \(y = mx + c\) or the point-slope form \(y - y_1 = m(x - x_1)\).Key characteristics of linear equations are:

• They graph as straight lines.
• The slope \(m\) represents the rate of change or how much \(y\) changes for a unit change in \(x\).
• They can have one solution, infinitely many solutions (if they describe the same line), or no solution at all (if they describe parallel lines).

Overall, mastering linear equations is crucial for understanding how various quantities relate to each other across different contexts, from daily life scenarios to complex scientific data.