Problem 18 - Elementary Algebra

Question

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ m=-6, y \text { -intercept }(0,-1) $$

Step-by-Step Solution

Answer

Answer: The slope-intercept form of the linear equation is y = -6x - 1.

Step 1: Identify the given values

We are given the following information about the line:$$m = -6, y\text{-intercept} = (0, -1)$$

Step 2: Write the slope-intercept equation of a line

Write the general form of the slope-intercept equation:$$y = mx + b$$

Step 3: Plug the given values into the equation

Since we are given the slope, m = -6, and the y-intercept, b = -1, we can substitute these values into the equation:$$y = -6x - 1$$

Step 4: Final equation of the line

The slope-intercept form of the linear equation with the given values is:$$y = -6x -1$$

Key Concepts

Linear Equation

A linear equation represents a straight line on a graph. It's one of the simplest forms of equations in mathematics and is frequently used to model relationships between variables. The standard form of a linear equation can be expressed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. However, there's another form called the slope-intercept form, which is particularly useful when you want to quickly determine the slope and y-intercept of a line.

The slope-intercept form is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This formula makes it easy to graph a line. You start by plotting the y-intercept on the y-axis and then use the slope to find another point. The simplicity and visual clarity of the slope-intercept form make it popular among students learning about linear equations.

Slope

The slope of a line measures how steep the line is. It is a crucial part of the linear equation in the slope-intercept form, represented by \(m\). The slope indicates the rate of change of the dependent variable (usually \(y\)) with respect to the independent variable (usually \(x\)).

The formula for the slope between two points, \((x_1, y_1)\) and \((x_2, y_2)\), is given by:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

A positive slope means that as \(x\) increases, \(y\) also increases, causing the line to go upwards. Conversely, a negative slope, like the one in our example with \(m = -6\), indicates that as \(x\) increases, \(y\) decreases, resulting in a downward-sloping line.
Understanding the slope is crucial as it helps us comprehend the relationship between variables in real-world scenarios, such as speed and time in physics.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis on a graph. In the slope-intercept form of a linear equation, \(y = mx + b\), the \(b\) value denotes the y-intercept. This point is significant because it reveals the value of \(y\) when \(x\) equals zero.

• It provides a starting point for plotting the graph of any linear equation.
• Helps determine how the line shifts vertically on the graph compared to other lines.

For instance, in our solution where \(b = -1\), the y-intercept is \((0, -1)\), meaning the line crosses the y-axis at -1. This makes it straightforward to plot the line on a graph since we already know one significant point the line passes through.
The y-intercept not only helps in graph plotting but also provides insights into the line's behavior without needing to calculate additional points.