Question

the pattern of squares continues. which equation below relates the number of squares (s) in a picture to the figure number (f)? (hint: apply substitution)
figure 1
figure 2
figure 3
\bigcirc f = s+7
\bigcirc s = s+6
\bigcirc s = f+6
\bigcirc f = s+6
which equation below corresponds to the table of values? *

Explanation:

Step1: Count squares in each figure

• Figure 1 (f=1): Let's count squares. The left and right columns have 3 each, middle has 1. Total: \(3 + 1 + 3 = 7\) squares. So \(s = 7\) when \(f = 1\).
• Figure 2 (f=2): Left and right 3 each, middle 2. Total: \(3 + 2 + 3 = 8\) squares. \(s = 8\) when \(f = 2\).
• Figure 3 (f=3): Left and right 3 each, middle 3. Total: \(3 + 3 + 3 = 9\) squares. \(s = 9\) when \(f = 3\).

Step2: Test each equation

• Option 1: \(f = s + 7\). For \(f=1\), \(s = 1 - 7 = -6\) (invalid, since squares can't be negative). Eliminate.
• Option 2: \(s = s + 6\). Subtract \(s\) from both sides: \(0 = 6\) (false). Eliminate.
• Option 3: \(s = f + 6\).
• For \(f=1\): \(s = 1 + 6 = 7\) (matches Figure 1).
• For \(f=2\): \(s = 2 + 6 = 8\) (matches Figure 2).
• For \(f=3\): \(s = 3 + 6 = 9\) (matches Figure 3).
• Option 4: \(f = s + 6\). For \(f=1\), \(s = 1 - 6 = -5\) (invalid). Eliminate.

Answer:

\(s = f + 6\) (the third option: \(s = f + 6\))