Question

for each expression, select all equivalent expressions from the list.
(a) ( x + x + x + x + x ) ( square , x + 5 ) ( square , 5x ) ( square , 5 + x ) ( square , 3x - 2x )
(b) ( 16y - 12 ) ( square , 4(4y - 3) ) ( square , 8y - 6 ) ( square , 4y ) ( square , 4 cdot 4y - 4 cdot 3 )

Explanation:

Part (a)

Step1: Simplify the original expression

The expression \( x + x + x + x + x \) has five \( x \) terms added together. By the definition of like - term addition, \( a\times x+a\times x+\cdots+a\times x \) (n times) is \( n\times x \). Here \( n = 5 \), so \( x + x + x + x + x=5x \).

Step2: Analyze each option

• Option \( x + 5 \): This is the sum of \( x \) and 5, which is not equivalent to \( 5x \).
• Option \( 5x \): As we calculated above, \( x + x + x + x + x = 5x \), so this is equivalent.
• Option \( 5 + x \): This is the sum of 5 and \( x \), which is not equivalent to \( 5x \).
• Option \( 3x-2x \): Simplify \( 3x - 2x=(3 - 2)x=x\), which is not equivalent to \( 5x \).

Part (b)

Step1: Analyze each option

• Option \( 4(4y - 3) \): Using the distributive property \( a(b - c)=ab - ac \), where \( a = 4 \), \( b = 4y \), \( c = 3 \). So \( 4(4y-3)=4\times4y-4\times3 = 16y-12 \), which is equivalent.
• Option \( 8y - 6 \): If we factor out 2 from \( 16y-12 \), we get \( 2(8y - 6) \), which is not equal to \( 8y - 6 \) (unless \( 2 = 1 \), which is false), so it is not equivalent.
• Option \( 4y \): This is a single - term expression with \( y \) coefficient 4, while \( 16y-12 \) has two terms, so it is not equivalent.
• Option \( 4\cdot4y-4\cdot3 \): By the order of operations and multiplication, \( 4\cdot4y = 16y \) and \( 4\cdot3 = 12 \), so \( 4\cdot4y-4\cdot3=16y - 12 \), which is equivalent.

Answer:

(a) The equivalent expression is \( 5x \) (the second option).
(b) The equivalent expressions are \( 4(4y - 3) \) (the first option) and \( 4\cdot4y-4\cdot3 \) (the fourth option).