Problem 64 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(57-64\), write an equation in the form \(y=m x+b\) of the line that is described. The line falls from left to right. It passes through the origin and a second point with opposite \(x\) - and \(y\) -coordinates.

Step-by-Step Solution

Answer

The equation of the given line is \(y = -x\).

Step 1: Determine the y-intercept

Given that the line passes through the origin, the y-intercept 'b' is 0 since the y-intercept is the point where the line crosses the y-axis, which happens to be the origin in this case.

Step 2: Calculate the slope

The slope 'm' of the line can be calculated using the formula m=(y2-y1)/(x2-x1). For the two points given (0,0) and (a,-a) the slope would be m=(-a-0)/(a-0) = -1.

Step 3: Construct the line equation

Now that the slope 'm' and y-intercept 'b' have been found, these can be substituted into the equation of a line y=m*x+b to get y = -x + 0 or simply y = -x.

Key Concepts

Equation of a Line

An equation of a line is a mathematical way to describe a straight line on a graph. It helps us understand the relationship between the variables \(x\) and \(y\). The most common form of a linear equation is \(y = mx + b\).
This equation is called the slope-intercept form. Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept. This form is favoured because it directly tells you the slope of the line and where it intersects the y-axis. Knowing the equation allows us to quickly understand key features of the line, like its direction and steepness.

Slope

The slope of a line is a measure of its steepness. It indicates how much \(y\) changes as \(x\) changes. In the equation \(y = mx + b\), \(m\) is the slope.

• The slope is calculated as \(\frac{(y_2-y_1)}{(x_2-x_1)}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.
• If the slope is positive, the line rises from left to right.
• If it is negative, the line falls from left to right, as in our exercise where the slope is \(-1\).

The larger the absolute value of the slope, the steeper the line. A slope of \(-1\) indicates a decrease of one unit in \(y\) for every unit increase in \(x\). This characteristic ensures predictability when interpreting graphs.

Y-intercept

The y-intercept is the point where the line crosses the y-axis. It is key to determining the vertical position of a line in a graph. In the slope-intercept form \(y = mx + b\), \(b\) is the y-intercept.
In our exercise, since the line passes through the origin, the y-intercept \(b\) is \(0\). This means the line will cross the y-axis at the point \((0, 0)\).
Knowing the y-intercept allows you to determine the starting point of the line when plotting it on a graph. It's especially useful because it provides direct proof of where the line begins its journey across the graph.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the backbone of equations and are used widely in mathematics to represent general relationships.
In the equation \(y = mx + b\), both \(mx\) and \(b\) are parts of an algebraic expression. Understanding them helps us manipulate and simplify equations effectively.

• \(mx\) shows the product of a constant \(m\) (the slope) and \(x\). It represents how much \(y\) will change concerning \(x\).
• \(b\) is the constant term, representing the y-intercept, the fixed value when \(x\) is \(0\).

These expressions allow us to transform a general relationship into specific, numerical forms, making them invaluable for solving problems and understanding mathematical connections.