Problem 28 - Prealgebra

Question

Comparing Tables Which of the tables below could be produced from the equation \(y=3 x+5 ?\) $$\begin{aligned} &a_{1}\\\ &=\\\ &\begin{array}{r|r} x & y \\ \hline-3 & -4 \\ \hline-2 & -7 \\ \hline-1 & -10 \\ \hline 0 & -10 \end{array} \end{aligned}$$ $$\begin{aligned} &\mathbf{b}\\\ &\begin{array}{r|r} x & y \\ \hline-3 & 14 \\ \hline-2 & 11 \\ \hline-1 & 8 \\ \hline 0 & 5 \end{array} \end{aligned}$$ $$\begin{aligned} &c\\\ &x\\\ &\begin{array}{c|c} x & y \\ \hline-3 & -4 \\ \hline-2 & -1 \\ \hline-1 & 2 \\ \hline 0 & 5 \end{array} \end{aligned}$$

Step-by-Step Solution

Answer

Table c satisfies the equation \(y = 3x + 5\).

Step 1: Understand the Given Equation

We are given the equation \(y=3x+5\). This represents a linear relationship between \(x\) and \(y\), where \(y\) changes based on the value of \(x\). Our task is to see which table contains values that satisfy this equation.

Step 2: Evaluate Table a

In table a, we check each \(x\) and \(y\) pair to see if they satisfy the equation \(y = 3x + 5\). First, for \(x = -3\), we calculate \(y = 3(-3) + 5 = -9 + 5 = -4\), which matches the table. Next, for \(x = -2\), we have \(y = 3(-2) + 5 = -6 + 5 = -1\), but the table shows \(y = -7\). The function does not match for \(x = -2\), so table a is not correct.

Step 3: Evaluate Table b

For table b, we again verify if each pair satisfies the equation. Start with \(x = -3\): \(y = 3(-3) + 5 = -9 + 5 = -4\), but the table shows \(y = 14\). For \(x = -2\), \(y = 3(-2) + 5 = -1\), but the table shows \(y = 11\). The values do not match the equation; thus, table b is incorrect.

Step 4: Evaluate Table c

Check table c values with the equation: For \(x = -3\), calculate \(y = 3(-3) + 5 = -4\) which matches. For \(x = -2\), \(y = 3(-2) + 5 = -1\), for \(x = -1\), \(y = 3(-1) + 5 = 2\), and for \(x = 0\), \(y = 3(0) + 5 = 5\). All values match the equation.

Step 5: Conclusion

After thorough checks, table c is the only one where all values of \(x\) and \(y\) satisfy the given equation \(y = 3x + 5\). Therefore, table c can be produced from the equation.

Key Concepts

Comparing Tables

When working with linear equations, it is often helpful to use tables to represent the relationship between two variables, typically referred to as \(x\) and \(y\). Each pair of \(x\) and \(y\) in a table should satisfy the given linear equation. To do this, substitute the \(x\) values into the equation to see if the resulting \(y\) values match those in the table.
Consider a linear equation like \(y = 3x + 5\). If you have a table of \(x\) values

• Calculate \(y\) for each \(x\) using the equation.
• Compare the calculated \(y\) values to the \(y\) values listed in the table.

If for every \(x\), the calculated \(y\) equals the \(y\) from the table, then the table correctly represents the equation. Understanding this concept helps in verifying if a table of values depicts the equation accurately or not.

Algebraic Expressions

Algebraic expressions form the backbone of linear equations. An algebraic expression is like a mathematical sentence that describes a relationship using numbers, variables, and operations. The equation \(y = 3x + 5\) combines the variable \(x\), the coefficient \(3\), and the constant \(5\).
Here's how to understand this structure:

• Variable: \(x\), which varies within the table and equation.
• Coefficient: \(3\), which indicates how much \(y\) changes for a unit change in \(x\).
• Constant: \(5\) is the \(y\)-intercept, where the line crosses the \(y\)-axis.

In essence, an algebraic expression provides a concise way of summarizing the relationship between variables. In this context, it dictates how to calculate \(y\) when given \(x\). Recognizing and understanding these parts in an equation will allow you to effectively work through problems involving linear relationships.

Linear Relationships

Linear relationships are characterized by a constant rate of change between two variables, depicted as a straight line graphically. This relationship is often described using a linear equation like \(y = 3x + 5\). Here is what defines it:

• Slope: The coefficient \(3\) represents the slope, describing how steep the line is.
• Intercept: The constant \(5\) is where the line crosses the \(y\)-axis, known as the \(y\)-intercept.

To identify a linear relationship from a table, check if the differences in \(y\) divided by differences in \(x\) remain constant. This constancy points to the steady rate of change depicted by a linear equation.
A solid grasp of linear relationships helps in recognizing consistent patterns and solving problems involving linear equations in various contexts. By understanding such relationships, you're better equipped to predict and analyze how changes in one variable affect another in a straightforward manner.