Problem 51 - Introductory Algebra for College Students

Question

find five solutions of each equation. Select integers for \(x,\) starting with \(-2\) and ending with \(2 .\) Organize your work in a table of values. $$y=-10 x$$

Step-by-Step Solution

Answer

The five pairs of solutions for the equation \( y = -10x \) are (-2, 20), (-1, 10), (0, 0), (1, -10), and (2, -20), which can be organized in a table of values.

Step 1: Understand the equation

The given equation is in the form \( y = ax \), which is a kind of simple linear equation where, 'a' is the coefficient of 'x' and it defines the steepness of the line.

Step 2: Select the integer values for \( x \)

We select five integer values for \( x \) starting from -2 and ending with 2. The five values are -2, -1, 0, 1, 2.

Step 3: Calculate the \( y \) values

Now for each value of \( x \), substitute it into the equation \( y = -10x \) to calculate the corresponding \( y \) value. This gives five pairs of values.

Step 4: Organize in a table of values

Once the \( y \) values are calculated, the pairs ( \( x, y \) ) can be organized in a table. The table has two columns, one for \( x \) and one for \( y \) . Each row corresponds to one solution of the equation.

Key Concepts

Understanding Linear Equations

Linear equations are fundamental in algebra and they form the basis for more complex mathematical concepts. A linear equation in one variable, like the one provided in the exercise, has the general form of
\( y = ax + b \), where
\( a \) and \( b \) are constants. The solution to such an equation is a set of points on a straight line when plotted on a graph. For the particular equation \( y = -10x \), there is no constant \( b \), which means that it's a special case of the linear equation where the line passes through the origin (0,0).
Each point on the line represented by the equation satisfies the equation, meaning if you substitute the coordinates of the point into the equation, both sides will be equal. Therefore, solving a linear equation typically means finding the set of all possible points that make the equation true.

Creating a Table of Values

A table of values is a systematic way to organize the solutions of an equation. It's especially useful for plotting graphs of equations or visualising how changing one variable affects another. For a linear equation such as \( y = -10x \), creating a table starts with choosing a range of values for \( x \). Typically, you would select values that are evenly spaced and include both positive and negative numbers. This approach demonstrates how both \( x \) and \( y \) change in response to each other.
The steps include choosing your \( x \) values, which in this case are -2, -1, 0, 1, and 2, and then using the equation to find the corresponding \( y \) values. These pairs of \( x \) and \( y \) values are then properly organized into two columns in the table, representing a clear relationship between the variables. This table can then be used to easily plot the points on a coordinate system.

Finding Integer Solutions

When we focus on finding integer solutions for linear equations, we limit our solutions to whole numbers. Integer solutions are particularly interesting because they represent exact points where the graph of a linear equation crosses grid lines on a traditional Cartesian plane. To find integer solutions for \( y = -10x \), like in the given exercise, it's helpful to select integer values for \( x \), which are then substituted into the equation to find the corresponding integer values for \( y \).
Integer solutions are easily identifiable on a graph and are helpful in various practical applications where fractional or decimal solutions may not be suitable or necessary. They provide exact numbers that can be used for counting, measuring, and other operations where precision is key. In this exercise, organizing integer solutions in a table not only simplifies the process of finding these specific points but also aids in visualizing the discrete nature of integer solutions on the graph of a linear equation.