Problem 49 - 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. $$y=12 x$$

Step-by-Step Solution

Answer

The table of values from \(x=-2\) to \(x=2\) is as follows: \n(-2,-24), \n(-1,-12), \n(0,0), \n(1,12), \n(2,24).

Step 1: Understand the given equation

The equation provided is a simple linear equation, \(y = 12x\). This means for every unit change in \(x\), the value of \(y\) changes by 12 times the amount.

Step 2: Initialize \(x\) with a starting value

For this exercise, the requested starting value for \(x\) is -2.

Step 3: Calculate value of \(y\)

Substitute \(x=-2\) into the equation. This yields \(y = 12*-2\), so \(y = -24\).

Step 4: Increase \(x\) by 1

Increase the value of \(x\) to -1.

Step 5: Repeat Step 3

Substitute \(x=-1\) into the equation. This yields \(y = 12*-1\), so \(y = -12\).

Step 6: Continue to increase \(x\)

Continue this process by incrementing \(x\) by 1, substituting the new value into the equation, and finding the corresponding value of \(y\). Do this until \(x\) reaches 2.

Step 7: Organize the results

Compile the results into a table listing the values of \(x\) and their corresponding values of \(y\).

Key Concepts

Table of Values

When working with linear equations like \(y = 12x\), it is helpful to organize your solutions using a table of values. A table of values helps us easily observe how changes in \(x\) affect \(y\). You start with a given range for \(x\), like from \(-2\) to \(2\), as mentioned in the exercise instructions.

To construct a table of values:

• Choose the integer values for \(x\) given in the problem.
• Substitute these values into the equation to find \(y\).
• Record each pair \((x, y)\) as a row in your table.

This creates a clear, visual representation of the linear relationship between \(x\) and \(y\). This method not only helps in solving the problem but also makes it easier to spot patterns or trends between the variables.

Integer Solutions

Integer solutions refer to the solutions of an equation where the resulting numbers are whole numbers. In our problem, the equation \(y = 12x\) is a linear equation, and we're calculating the solutions for specific integer values of \(x\) (starting at \(-2\) and ending at \(2\)).

Here’s how to find integer solutions for the given problem:

• Substitute integer values into the equation, one at a time.
• Compute the corresponding \(y\) value by performing the necessary multiplication.
• Ensure the computed \(y\) values are also integers, which in this exercise they will be, as multiplying integers yields another integer.

For example, substituting \(x = -2\) into the equation yields \(y = 12 \times (-2) = -24\), an integer solution, continuing this way for each subsequent \(x\) value ensures each \((x, y)\) pair is an integer solution.

Algebraic Expressions

An algebraic expression involves variables, numbers, and operations, forming a significant part of algebra. In the expression \(y = 12x\), \(y\) and \(x\) are the variables and \(12x\) is the algebraic expression.

Understanding the role of each component:

• Variables: In this case, \(x\) and \(y\) represent unknown quantities that we determine using the equation.
• Coefficient: The number \(12\) is the coefficient, which indicates how much \(x\) influences \(y\).
• Operation: Multiplication, in this expression, indicates that \(y\) is a result of \(x\) being multiplied by \(12\).

Algebraic expressions form the foundation for understanding linear equations. When we substitute a value for \(x\), the expression \(12x\) determines the specific value of \(y\). Comprehending these basics makes solving equations straightforward.