Problem 54 - 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. $$y=6 x-4$$

Step-by-Step Solution

Answer

The values of \(y\) for \(x\) ranging from \(-2\) to \(2\) are \(-16, -10, -4, 2\), and \(8\) respectively.

Step 1: Substitute \(x=-2\)

Replace \(x\) with \(-2\) in the equation, it becomes \(y=6(-2)-4\) which simplifies to \(y=-12-4=-16\)

Step 2: Substitute \(x=-1\)

Next, replace \(x\) with \(-1\) in the equation, it becomes \(y=6(-1)-4\) which simplifies to \(y=-6-4=-10\)

Step 3: Substitute \(x=0\)

Then, replace \(x\) with \(0\) in the equation, it becomes \(y=6(0)-4\) which simplifies to \(y=0-4=-4\)

Step 4: Substitute \(x=1\)

After that, replace \(x\) with \(1\) in the equation, it becomes \(y=6(1)-4\) which simplifies to \(y=6-4=2\)

Step 5: Substitute \(x=2\)

Finally, replace \(x\) with \(2\) in the equation, it becomes \(y=6(2)-4\) which simplifies to \(y=12-4=8\)

Step 6: Listing down in tabular form

Now, we list down the \(x\) values along with their corresponding \(y\) values in a table. The table is arranged as follows:\n \n|x|y|\n|---|---|\n|-2|-16|\n|-1|-10|\n|0|-4|\n|1|2|\n|2|8|

Key Concepts

Table of Values

Using a table of values is a great way to visually organize the solutions to a linear equation. Think of it as a quick way to see how different input values influence the output of your equation. When dealing with linear equations like \( y = 6x - 4 \), making a table can help identify patterns and relationships between the variables.
Steps to create a table of values include:

• Choosing a range of integer values for \( x \). In our example, we selected integers from \( -2 \) to \( 2 \).
• Substituting each of these \( x \) values into the equation to solve for \( y \).
• Recording each pair \((x, y)\) in your table.

As you calculate, you'll fill out the table and see how \( y \) changes. This method is especially helpful for initial exploration of equations and relationships, offering a straightforward snapshot of data points.

Integer Solutions

Integer solutions are solutions to equations where the answer is a whole number. They are key to this type of problem because we are asked to calculate solutions for the integer values of \( x \). Whole numbers are easy to work with and avoid complications with fractions or decimals.
When we solve for \( y \) in the equation \( y = 6x - 4 \) using integer values of \( x \), each computed \( y \) remains an integer as well. This simplifies graphing and interpretation.

• For \( x = -2 \), \( y = -16 \)
• For \( x = -1 \), \( y = -10 \)
• For \( x = 0 \), \( y = -4 \)
• For \( x = 1 \), \( y = 2 \)
• For \( x = 2 \), \( y = 8 \)

These integer pairs are easy to plot on a grid or graph, which can help visualize the linear relationship.

Substitution Method

The substitution method is a simple and direct way to find the corresponding \( y \) value for each chosen \( x \) value. It involves replacing the variable \( x \) in the linear equation with a specific number and solving for \( y \).
Here's how it is applied step-by-step:

• Start with a known value of \( x \). Let's use \( x = -2 \) as a starting point.
• Substitute \( -2 \) for \( x \) in the equation \( y = 6x - 4 \). The equation becomes \( y = 6(-2) - 4 \).
• Perform the calculation. Here, \( 6(-2) = -12 \) first, then subtract \( 4 \) to get \( y = -16 \).
• Repeat this process for other \( x \) values.

Using substitution, you can quickly determine multiple solutions and construct a complete table of values. It is a logical and repeatable process that facilitates understanding of the linear relationship described by the equation.