Problem 18 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line described below. $$ m=-4, b=-\frac{3}{7} $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = -4x - \frac{3}{7}\).

Step 1: Understanding the Slope-Intercept Form of a Line

The slope-intercept form for linear equations is \(y = mx + b\), where \(m\) stands for the slope of the line, and \(b\) stands for the y-intercept of the line. Thus, we will use this formula to create an equation for the line.

Step 2: Substitution of values

Given \(m = -4\) and \(b = -3/7\), the values need to be substituted into the equation from Step 1.

Step 3: Formulate the Equation of the Line

Substituting the given \(m\) and \(b\) values into the slope-intercept equation \(y = mx + b\) we get \(y = -4x - \frac{3}{7}\). Thus, this is the equation of the line in slope-intercept form.

Key Concepts

Linear Equations

Linear equations describe a relationship between two variables, typically represented by \(x\) and \(y\). These equations graph as straight lines on a coordinate plane. The general form of a linear equation is \(y = mx + b\), where \(m\) denotes the slope and \(b\) denotes the y-intercept.
Linear equations can represent data points showing a linear trend or model simple relationships in real-world scenarios. Some key points include:

• They are first-degree equations, meaning the highest exponent of the variable is one.
• They form straight lines when graphed.
• The solution of a linear equation is the point(s) at which the line crosses the axes.

Understanding these properties is crucial, as they help in modeling and solving real-life problems.

Slope of a Line

The slope of a line is a measure of how steep the line is. It indicates the rate of change between the two variables in the linear equation.
A positive slope means the line ascends from the left to right, while a negative slope means it descends.
The slope \(m\) is calculated as the "rise" over "run", or the change in \(y\) over the change in \(x\) between two points. In mathematics, this can be expressed as: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] where \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line.
It's important to understand the slope because it determines the direction and angle of the line. For instance:

• A slope of 0 indicates a horizontal line, showing no change in \(y\) as \(x\) changes.
• An undefined slope suggests a vertical line, which occurs without any change in \(x\) as \(y\) changes.

The slope is critical in real-life applications, such as calculating speeds or predicting trends.

Y-Intercept

The y-intercept is the point where the line intersects the y-axis on a graph. It represents the value of \(y\) when \(x\) is zero.
In the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\).
Knowing the y-intercept is useful as it gives insight into the starting point of the function when graphed.

• If \(b\) is positive, the line crosses the y-axis above the origin.
• If \(b\) is negative, the line crosses below the origin.
• If \(b\) is zero, the line goes through the origin itself.

In real-world contexts, the y-intercept can represent initial conditions or starting values, helping to predict and understand behavior across a variety of disciplines.