Problem 27 - Prealgebra

Question

Comparing Tables which of the tables below could be produced from the equation \(y=2 x-6 ?\) $$\begin{aligned} &\mathbf{a}\\\ &\begin{array}{c|c} x & y \\ \hline 0 & 6 \\ \hline 1 & 4 \\ \hline 2 & 2 \\ \hline 3 & 0 \\ \hline \end{array} \end{aligned}$$ b. \begin{tabular}{|c|c|} \hline\(x\) & \(y\) \\ \hline 0 & \(-6\) \\ \hline 1 & \(-4\) \\ \hline 2 & \(-2\) \\ \hline 3 & 0 \\ \hline \end{tabular} \begin{aligned} &c\\\ &\begin{array}{c|c} x & y \\ \hline 0 & -6 \\ \hline 1 & -5 \\ \hline 2 & -4 \\ \hline 3 & -3 \\ \hline \end{array} \end{aligned}

Step-by-Step Solution

Answer

The equation can produce Table b.

Step 1: Understand the Equation

The given equation is \( y = 2x - 6 \). This is a linear equation with a slope of 2 and a y-intercept of -6.

Step 2: Plug in Values for Table a

For Table a, substitute the x-values (0, 1, 2, 3) into the equation to find the corresponding y-values: - For \( x = 0 \), \( y = 2(0) - 6 = -6 \)- For \( x = 1 \), \( y = 2(1) - 6 = -4 \)- For \( x = 2 \), \( y = 2(2) - 6 = -2 \)- For \( x = 3 \), \( y = 2(3) - 6 = 0 \).None of these y-values match Table a, so Table a cannot be produced by the equation.

Step 3: Plug in Values for Table b

For Table b, substitute the x-values (0, 1, 2, 3) into the equation to find the corresponding y-values:- For \( x = 0 \), \( y = 2(0) - 6 = -6 \)- For \( x = 1 \), \( y = 2(1) - 6 = -4 \)- For \( x = 2 \), \( y = 2(2) - 6 = -2 \)- For \( x = 3 \), \( y = 2(3) - 6 = 0 \).These y-values match exactly with Table b, so Table b could be produced by the equation.

Step 4: Plug in Values for Table c

For Table c, substitute the x-values (0, 1, 2, 3) into the equation to find the corresponding y-values:- For \( x = 0 \), \( y = 2(0) - 6 = -6 \)- For \( x = 1 \), \( y = 2(1) - 6 = -4 \), but Table c shows -5 instead.- For \( x = 2 \), \( y = 2(2) - 6 = -2 \), but Table c shows -4 instead.- For \( x = 3 \), \( y = 2(3) - 6 = 0 \), but Table c shows -3 instead.Since none of these y-values besides the first match Table c, it cannot be produced by the equation.

Key Concepts

Tables in Math

When solving linear equations, tables in math can be extremely helpful. A table organizes values, showing the relationship between variables. In our example, we have the equation \( y = 2x - 6 \). Here, \( x \) is an independent variable, and \( y \) is the dependent variable. Each row in our table represents one set of \( x \) and \( y \) values, illustrating how they relate through the equation.
Using a table helps you visualize the changes in \( y \) as \( x \) increases or decreases. You can test different values of \( x \) to see what \( y \) equals, and whether or not these values might align with any given tables, as in this exercise. In this way, tables serve as a comparative tool, confirming whether a table aligns with the provided equation or not.

Slope and Y-Intercept

The slope and y-intercept are crucial parts of understanding a linear equation like \( y = 2x - 6 \). The slope describes how steep the line is, and it tells us the rate at which \( y \) changes as \( x \) changes. In this equation, the slope is \( 2 \), meaning for every increase of \( 1 \) unit in \( x \), \( y \) increases by \( 2 \) units.
The y-intercept is the value of \( y \) where the line crosses the y-axis (where \( x = 0 \)). For this equation, the y-intercept is \(-6\). It tells us that even when \( x \) is 0, \( y \) starts at \(-6\). Knowing the slope and y-intercept lets you quickly sketch the line of the equation and predict the behavior of \( y \) given any \( x \). This helps in determining which table accurately represents the solution of the equation.

Evaluating Equations

Evaluating equations involves substituting the x-values into the equation to find the y-values. This is key to comparing which tables match the equation, as seen in our exercise. To evaluate \( y = 2x - 6 \), you take each x-value from the table and plug it into the equation:

• For \( x = 0 \), substitute to get \( y = 2(0) - 6 = -6 \).
• For \( x = 1 \), substitute to get \( y = 2(1) - 6 = -4 \).
• Proceed with \( x = 2 \) and \( 3 \) similarly.

By calculating the y-values this way and comparing them to those in the tables, you can confirm if the table represents the solution to the equation accurately. This practice reinforces understanding of functional relationships and aids in solving similar problems.