Question

If \(y - x = 2\) and \(y - z = 3\), which of the following best represents the relative positions of x, y, and z on the number line? (Note: The figures are drawn to scale.)
\includegraphics[width=0.35\linewidth]{19.png}

Hint:

A quick way to solve this is to pick a value for one variable and solve for the others. For example, let \(y=5\). Then \(5 - x = 2 \Rightarrow x = 3\), and \(5 - z = 3 \Rightarrow z = 2\). The numbers are \(z=2, x=3, y=5\), confirming the order \(z<x<y\).

Answer 1

Step 1: Understanding the Concept:
The given equations describe the distances and relative ordering between three points on a number line. We need to combine these pieces of information to determine the overall order of \(x, y,\) and \(z\).
Step 2: Key Formula or Approach:
Analyze each equation to determine the relationship between the two variables involved. An equation of the form \(a - b = k\) where \(k\) is positive implies \(a>b\).
Step 3: Detailed Explanation:
Let's analyze the first equation:
\[ y - x = 2 \] Since the result is positive, this means \(y>x\). We can rewrite this as \(y = x + 2\), which tells us that \(y\) is located 2 units to the right of \(x\) on the number line.
Now let's analyze the second equation:
\[ y - z = 3 \] Since the result is positive, this means \(y>z\). We can rewrite this as \(y = z + 3\), which tells us that \(y\) is located 3 units to the right of \(z\) on the number line.
So far, we know that \(y\) is the greatest of the three numbers. Now we need to compare \(x\) and \(z\). We can express both in terms of \(y\):
From the first equation: \(x = y - 2\)
From the second equation: \(z = y - 3\)
To find \(x\), we subtract 2 from \(y\). To find \(z\), we subtract 3 from \(y\). Since we are subtracting a larger number from \(y\) to get \(z\), \(z\) must be smaller than \(x\).
Thus, \(z<x\).
Combining all the information, we get the final order: \(z<x<y\).
Step 4: Final Answer:
The correct order of the points on the number line from left to right is \(z\), then \(x\), then \(y\). This corresponds to the arrangement shown in option (C).