Question

(a) select all equivalent expressions from the list for the expression. x+x+x+x 2x - 2x 4x 4 + x x + 4 (b) select all equivalent expressions from the list for the expression. 72y - 40 8·9y - 8·5 36y - 20 32y 8(9y - 5)

Explanation:

Part (a)

Step1: Simplify the original expression

The original expression is \( x + x + x + x \). Combining like terms (adding the coefficients of \( x \), where each \( x \) has a coefficient of 1), we have \( 1x+1x + 1x+1x=(1 + 1+1 + 1)x=4x \).

Step2: Analyze each option

• For \( 2x-2x \): Simplifying this, we get \( 0 \), which is not equal to \( 4x \) (unless \( x = 0 \), but we need an equivalent expression for all \( x \)).
• For \( 4x \): This is the same as our simplified original expression.
• For \( 4 + x \): This is a sum of a constant and a variable, not equivalent to \( 4x \) (e.g., if \( x = 2 \), \( 4x=8 \) and \( 4 + x = 6 \)).
• For \( x + 4 \): Similar to the above, it's a variable and a constant, not equivalent to \( 4x \).

Part (b)

Step1: Simplify the original expression or factor it

The original expression is \( 72y-40 \). We can factor out the greatest common factor (GCF) of 72 and 40. The GCF of 72 and 40 is 8. Factoring out 8, we get \( 8(9y - 5) \), which is the same as \( 8\cdot9y-8\cdot5 \) (by the distributive property \( a(b - c)=ab - ac \), here \( a = 8 \), \( b=9y \), \( c = 5 \)).

Step2: Analyze each option

• For \( 8\cdot9y-8\cdot5 \): This is exactly the expanded form of \( 8(9y - 5) \), which is equivalent to \( 72y-40 \) (since \( 8\times9y=72y \) and \( 8\times5 = 40 \)).
• For \( 36y-20 \): If we factor out 2 from \( 72y-40 \), we get \( 2(36y - 20) \), but \( 36y-20\) is not equivalent to \( 72y - 40 \) (e.g., if \( y = 1 \), \( 72y-40=32 \) and \( 36y - 20 = 16 \)).
• For \( 32y \): This is a single - term expression, while \( 72y-40 \) is a two - term expression. They are not equivalent (e.g., \( y = 1 \), \( 72y-40 = 32 \), \( 32y=32 \) when \( y = 1 \), but if \( y = 2 \), \( 72y-40=144 - 40 = 104 \) and \( 32y = 64 \)).
• For \( 8(9y - 5) \): This is the factored form of \( 72y-40 \) (using the distributive property in reverse), so it is equivalent.

Answer:

(a) The equivalent expression is \( 4x \). So we select the option \( 4x \).

(b) The equivalent expressions are \( 8\cdot9y - 8\cdot5 \) and \( 8(9y - 5) \). So we select the options \( 8\cdot9y - 8\cdot5 \) and \( 8(9y - 5) \).