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

Step-by-Step Solution

Answer

The five solutions to the equation \(y=6x-4\) for integer values of \(x\) from -2 to 2 are {(-2, -16), (-1, -10), (0, -4), (1, 2), (2, 8)}.

Step 1: Start with \(x=-2\)

To begin with, substitute -2 in for \(x\) in the given equation to get the corresponding \(y\)-value: \(y = 6*-2 - 4 = -12 -4 = -16\).

Step 2: Continue with \(x=-1\)

Now try substituting -1 for \(x\) in the original equation. Calculate \(y = 6*-1 - 4 = -6 -4 = -10\).

Step 3: Substitute \(x=0\)

Replace \(x\) with 0 in the equation. Then, compute \(y = 6*0 - 4 = 0 - 4 = -4\)

Step 4: Proceed with \(x=1\)

Considering \(x=1\), put it into the equation and calculate \(y = 6*1 - 4 = 6 -4 = 2\).

Step 5: End with \(x=2\)

Lastly, use \(x=2\) in the equation, and work out \(y = 6*2 - 4 = 12 -4 = 8\).

Key Concepts

Table of Values

Creating a table of values is an organized way to determine the solutions of an equation for specific values of a variable. In this exercise, the equation is given as \( y = 6x - 4 \). A table of values helps in visualizing how the dependent variable \( y \) changes as the independent variable \( x \) is altered.
To start, we designate integer values for \( x \) from \(-2\) to \(2\). When these values are substituted into the equation, we calculate the corresponding \( y \)-values.

This approach lets you see the relationship between \( x \) and \( y \) clearly:

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

Having these pairs organized in a table allows quick referencing and a structured method of handling equations.

Integer Solutions

Integer solutions refer to those solutions of an equation where both the \( x \) and \( y \) values are integers. Finding integer solutions is often essential when graphing equations, especially in classroom settings where simplicity is key.
Let's look at our equation, \( y = 6x - 4 \). For each selected integer \( x \) value (from \(-2\) to \(2\)), we calculated the corresponding integer \( y \) values:

• 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 represent our integer solutions in this context, highlighting the sequence between input values and their resulting outputs. When graphed, they illustrate a straight line, since this is a linear equation.

Coordinate Pairs

In mathematics, coordinate pairs are used to denote the position of a point on a graph, typically represented as \( (x, y) \). For the given equation \( y = 6x - 4 \), each integer solution describes a precise location on the coordinate plane.
By substituting the same integer values for \( x \) from \(-2\) to \(2\) and calculating \( y \), we form the following coordinate pairs:

• \((-2, -16)\)
• \((-1, -10)\)
• \((0, -4)\)
• \((1, 2)\)
• \((2, 8)\)

These pairs can be plotted as points on a Cartesian plane. Connecting these points reveals the graph of the linear equation, offering a visual representation of the relationship. It simplifies the understanding of how changes in \( x \) affect \( y \). Therefore, coordinate pairs are crucial for graphing and interpreting linear relationships.