Problem 56 - 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. $$y=-5 x+9$$

Step-by-Step Solution

Answer

The corresponding y-values for x-values from -2 to 2 in the equation \(y = -5x + 9\) are 19, 14, 9, 4, -1 respectively. The solutions can be organized in a table as follows: \[\begin{align*} x & y \ \ -2 & 19 \ \ -1 & 14 \ \ 0 & 9 \ \ 1 & 4 \ \ 2 & -1 \ \end{align*}.\]

Step 1: Understand the Equation

We are given the linear equation \(y = -5x + 9\). This equation is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept.

Step 2: Substitute \(x\) values

We need to find the corresponding \(y\) values for the integer \(x\) values from -2 to 2. We will substitute these \(x\) values in the equation and calculate the resulting \(y\) values respectively. For \(x = -2\), \(y = -5(-2) + 9 = 10 + 9 = 19\). For \(x = -1\), \(y = -5(-1) + 9 = 5 + 9 = 14\). For \(x = 0\), \(y = -5(0) + 9 = 0 + 9 = 9\). For \(x = 1\), \(y = -5(1) + 9 = -5 + 9 = 4\). For \(x = 2\), \(y = -5(2) + 9 = -10 + 9 = -1\).

Step 3: Organize in a Table

After calculating the \(y\) values, we organize the solutions as a table of x and y values. The table is as follows: \[\begin{align*} x & y \ \ -2 & 19 \ \ -1 & 14 \ \ 0 & 9 \ \ 1 & 4 \ \ 2 & -1 \ \end{align*}\]

Key Concepts

Understanding Slope-Intercept Form

The slope-intercept form of a linear equation is a powerful way to express a line on a coordinate plane. It usually looks like this: \( y = mx + b \). Here, \( m \) represents the slope of the line, while \( b \) tells us where the line intersects the y-axis, known as the y-intercept. This form allows you to quickly understand and graph a line.

Let's break down what these terms mean.

• **Slope (\( m \))**: The slope is a measure of how steep the line is. It's calculated as the "rise" (change in y) over the "run" (change in x). A positive slope means the line goes up as it moves from left to right, while a negative slope means it moves down.
• **Y-Intercept (\( b \))**: This is where the line crosses the y-axis. It gives you a starting point to draw the line from. For example, in our equation \( y = -5x + 9 \), the y-intercept is 9, meaning the line crosses the y-axis at point (0,9).

Understanding these concepts makes it easier to visualize the line without plotting every single point.

Creating a Table of Values

A table of values is a simple method to find and organize solutions for a linear equation. It's especially helpful when you need to find multiple points to graph a line.

The process involves choosing values for \( x \) and computing the corresponding \( y \) values using the equation. For our example, let's pick \( x \) values from -2 to 2 and compute their respective \( y \) values:

• For \( x = -2 \), \( y = -5(-2) + 9 = 19 \).
• For \( x = -1 \), \( y = -5(-1) + 9 = 14 \).
• For \( x = 0 \), \( y = -5(0) + 9 = 9 \).
• For \( x = 1 \), \( y = -5(1) + 9 = 4 \).
• For \( x = 2 \), \( y = -5(2) + 9 = -1 \).

A table is an organized way of presenting these values, making it easier to see the pattern and trend of the line. Each pair \((x, y)\) can be plotted as a point on a Cartesian plane.

Finding Integer Solutions

Integer solutions are simply the solutions where both \( x \) and \( y \) are integers. In our linear equation \( y = -5x + 9 \), we seek integer solutions that satisfy this equation.

For practical applications, integer solutions are essential as they connect to real-world situations where partial values don't make sense, like counting objects or people.

By selecting integer \( x \) values such as \(-2, -1, 0, 1,\) and \( 2 \) and using our linear equation, we derived integer \( y \) values such as 19, 14, 9, 4, and -1. The goal is to have both inputs and outputs as whole numbers. This concept is particularly useful in scenarios where precise, countable outcomes are needed, like inventory counts or seating arrangements.