Problem 51 - Introductory and Intermediate Algebra for College Students 4th

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 solutions organized in a table of values are: \((-2, 20), (-1, 10), (0, 0), (1, -10), (2, -20)\)

Step 1: Values Selection

Start by choosing the required values for \(x\) that are integers from \(-2\) to \(2\). This gives \(-2, -1, 0, 1, 2\).

Step 2: Substitute Values and Calculate

Substitute these \(x\) values into equation \(y=-10x\) and solve to find correspondig values of \(y\). Processing each value one-by-one: when \(x=-2\), \(y=-10*(-2)=20\); when \(x=-1\), \(y=-10*(-1)=10\); when \(x=0\), \(y=-10*0=0\); when \(x=1\), \(y=-10*1=-10\); and when \(x=2\), \(y=-10*2=-20\).

Step 3: Organize Values in a Table

Create a table of values to organize the results. The table will have two columns, one for \(x\) and one for \(y\). The rows will contain the values for each \(x\) and the corresponding \(y\), obtained from the equation. For this problem, the table appears as follows: \n \[ \begin{array}{cc} X & Y \ -2 & 20 \ -1 & 10 \ 0 & 0 \ 1 & -10 \ 2 & -20 \end{array} \]

Key Concepts

Table of Values

The table of values is a pivotal tool in understanding how different values of one variable affect another in an equation. To create a table of values for the equation \( y = -10x \), begin by selecting a range of integer values for \( x \), which in our current example are \( -2, -1, 0, 1, \) and \( 2 \).

After picking the values for \( x \), calculate the corresponding \( y \) values by substituting each \( x \) into the equation.

• For \( x = -2 \): \( y = -10(-2) = 20 \)
• For \( x = -1 \): \( y = -10(-1) = 10 \)
• For \( x = 0 \): \( y = -10(0) = 0 \)
• For \( x = 1 \): \( y = -10(1) = -10 \)
• For \( x = 2 \): \( y = -10(2) = -20 \)

These calculations are neatly organized in a table with two columns—one for \( x \) and one for \( y \). Each row houses an \( x \) value paired with its calculated \( y \) value, providing a clear visual representation of the relationship between the variables.

Substitution Method

The substitution method is a fundamental technique used to solve equations. It involves replacing a variable with its value to find the corresponding output. In the case of the linear equation \( y = -10x \), you substitute the chosen integer values for \( x \) to find the corresponding values for \( y \).

The process is straightforward: take a chosen value for \( x \), insert it into the equation in place of \( x \), perform the multiplication, and solve for \( y \). For example, using the value \( x = 2 \), substitution gives us \( y = -10 \times 2 = -20 \). This method is particularly useful because it can be applied to any function or equation to find specific points, and it lays the groundwork for creating a graph of the equation.

Linear Equation Graphing

Once you have a set of solutions organized in a table of values, graphing the linear equation becomes a task of plotting and connecting points. Each row of the table provides coordinates for a point on the graph. For instance, the point (2, -20) suggests that when the horizontal axis (usually representing \( x \)) has a value of 2, the vertical axis (representing \( y \)) will be at -20.

To graph the equation \( y = -10x \), start by plotting the points given by your table on a coordinate grid. Next, draw a straight line through these points, as the graph of a linear equation will always be a straight line. The line visualizes how \( y \) changes with \( x \), which is especially helpful in identifying patterns, trends, and the slope of the relationship. In this case, the slope is -10, indicating a steep decline of \( y \) as \( x \) increases.