Question

problem 3
in the diagram shown, x represents the weight of each triangle, and 3 represents the weigh of each square.
select all the equations that could represent a balanced hanger.
□ 6x = 6
□ x + 9 = 3x + 3
□ 6 = 2x
□ 6 + 12x = 12 + 6x
□ 2x + 12 = 4x + 6

Explanation:

Step1: Analyze left side weight

Left side: 2 triangles (weight \(2x\)) + 4 squares (each 3, so \(4\times3 = 12\)). Total: \(2x + 12\).

Step2: Analyze right side weight

Right side: 4 triangles (weight \(4x\)) + 2 squares (each 3, so \(2\times3 = 6\)). Total: \(4x + 6\).

Step3: Check each equation

• \(6x = 6\): Left total \(2x + 12\), right \(4x + 6\). \(6x=6\) implies \(x=1\), but \(2(1)+12=14\), \(4(1)+6=10\) (not equal). Wrong.
• \(x + 9 = 3x + 3\): Simplify to \(2x=6\), \(x=3\). Left: \(2(3)+12=18\), right: \(4(3)+6=18\) (equal). Correct.
• \(6 = 2x\): \(x=3\). Left: \(2(3)+12=18\), right: \(4(3)+6=18\) (equal). Correct.
• \(6 + 12x = 12 + 6x\): Simplify to \(6x=6\), \(x=1\) (wrong as above).
• \(2x + 12 = 4x + 6\): Simplify to \(2x=6\), \(x=3\). Left and right equal. Correct.

Answer:

• \(x + 9 = 3x + 3\)
• \(6 = 2x\)
• \(2x + 12 = 4x + 6\)