Question

real numbers
identifying equivalent algebraic expressions
for each expression, select all equivalent expressions from the list.
(a) $8(5x - 2)$
$\square$ $13x - 2$
$\square$ $8\cdot 5x - 8\cdot 2$
$\square$ $40x - 2$
$\square$ $40x - 16$
(b) $10y + 2y + 10y$
$\square$ $22y$
$\square$ $22 + y$
$\square$ $10y + 12y$
$\square$ $12y - 10y$

Explanation:

Part (a): Analyze \( 8(5x - 2) \)

Step 1: Apply Distributive Property

The distributive property states that \( a(b - c)=ab - ac \). For \( 8(5x - 2) \), we multiply 8 by each term inside the parentheses:
\( 8\times5x - 8\times2 = 40x - 16 \)

Step 2: Simplify \( 8\cdot5x - 8\cdot2 \)

Calculate \( 8\times5x = 40x \) and \( 8\times2 = 16 \), so \( 8\cdot5x - 8\cdot2=40x - 16 \), which is the same as the result from Step 1.

Step 3: Check Other Expressions

• \( 40x - 2 \): This is not equivalent because we should have \( - 16 \) (from \( 8\times2 \)) not \( - 2 \).
• \( 13x - 2 \): \( 8\times5x=40x

eq13x \), so not equivalent.

Part (b): Analyze \( 10y + 2y+10y \)

Step 1: Combine Like Terms

Like terms are terms with the same variable. Combine \( 10y + 2y+10y \):
\( (10y + 10y)+2y=20y + 2y = 22y \) or \( 10y+2y + 10y=(10 + 2+10)y=22y \)

Step 2: Check Other Expressions

• \( 10y + 12y \): \( 10y+12y = 22y\)? \( 10 + 12 = 22 \), so \( 10y+12y=22y \) (wait, no, original is \( 10y + 2y+10y \). Wait, let's re - evaluate. Wait, \( 10y+2y + 10y=(10 + 2+10)y = 22y \). \( 10y+12y=(10 + 12)y = 22y \)? Wait, no, \( 10+12 = 22 \), but the original has \( 10y+2y + 10y \). Wait, maybe a typo in the problem? Wait, no, let's check again. Wait, \( 10y+2y+10y=22y \). Let's check the options:
• \( 22y \): Equivalent, as shown.
• \( 22 + y \): Not equivalent, it's a constant plus a variable, not a single - variable term.
• \( 12y - 10y=2y

eq22y \)

• \( 10y + 12y = 22y\)? Wait, \( 10y+12y=(10 + 12)y = 22y \), but the original is \( 10y+2y + 10y \). Wait, maybe the original expression was \( 10y+2y + 10y \), and \( 10y + 12y \): Wait, \( 10y+2y+10y = 22y \), and \( 10y + 12y=22y \) (since \( 10 + 12 = 22 \)). Wait, also, \( 12y-10y = 2y

eq22y \), \( 10y + 12y = 22y \), \( 22 + y
eq22y \), \( 22y \) is equivalent, and \( 10y + 12y \) is also equivalent? Wait, no, \( 10y+2y + 10y=22y \), and \( 10y + 12y = 22y \) (because \( 10 + 12 = 22 \)). Wait, maybe I made a mistake. Wait, \( 10y+2y+10y=(10 + 2+10)y = 22y \), \( 10y+12y=(10 + 12)y = 22y \), so both \( 22y \) and \( 10y + 12y \) are equivalent? Wait, no, \( 10y+2y+10y = 22y \), and \( 10y + 12y=22y \), so they are equivalent. But let's go back to the problem.

Wait, for part (a), the equivalent expressions for \( 8(5x - 2) \) are \( 40x - 16 \) and \( 8\cdot5x - 8\cdot2 \) (which is \( 40x - 16 \)). For part (b), the equivalent expressions for \( 10y + 2y+10y \) are \( 22y \) and \( 10y + 12y \) (since \( 10y+2y + 10y=22y \) and \( 10y + 12y = 22y \))? Wait, no, \( 10y+2y+10y=(10 + 2+10)y = 22y \), \( 10y+12y=(10 + 12)y = 22y \), so yes. But let's check the original options again.

For part (a) options: \( 8(5x - 2) \)

• Equivalent: \( 40x - 16 \), \( 8\cdot5x - 8\cdot2 \)
• Not equivalent: \( 40x - 2 \), \( 13x - 2 \)

For part (b) options: \( 10y + 2y+10y \)

• Equivalent: \( 22y \), \( 10y + 12y \)
• Not equivalent: \( 22 + y \), \( 12y - 10y \)

Answer:

Part (a) Equivalent Expressions:

• \( 40x - 16 \)
• \( 8\cdot5x - 8\cdot2 \)

Part (b) Equivalent Expressions:

• \( 22y \)
• \( 10y + 12y \)