Question

real numbers
identifying equivalent algebraic expressions
for each expression, select all equivalent expressions from the list.
(a) $7x - 4x + 2x$
$\square$ $3x + 2x$
$\square$ $5 + x$
$\square$ $3x - 2x$
$\square$ $5x$
(b) $3(1 + 7y)$
$\square$ $3 + 1 \cdot 3 + 7y$
$\square$ $3 + 21y$
$\square$ $3 \cdot 1 + 1 + 3 \cdot 7y$
$\square$ $3 + 7y$

Explanation:

Step1: Simplify \(7x - 4x + 2x\)

Combine like terms: \(7x - 4x = 3x\), then \(3x + 2x = 5x\). Also, \(3x + 2x = 5x\) directly, and \(7x - 4x + 2x=(7 - 4 + 2)x = 5x\), \(3x - 2x=x\) (no), \(5 + x\) (no), \(3x + 2x = 5x\) (yes), \(5x\) (yes).

Step2: Simplify \(3(1 + 7y)\)

Use distributive property: \(3\times1+3\times7y = 3 + 21y\). Check others: \(3 + 1\cdot3 + 7y=3 + 3 + 7y = 6 + 7y\) (no), \(3\cdot1 + 1 + 3\cdot7y=3 + 1 + 21y = 4 + 21y\) (no), \(3 + 7y\) (no), \(3 + 21y\) (yes).

Answer:

(a) Equivalent to \(7x - 4x + 2x\): \(3x + 2x\), \(5x\)
(b) Equivalent to \(3(1 + 7y)\): \(3 + 21y\)