Question

(a) select all equivalent expressions from the list for the expression.
$9x + 63$
$\square\\ 9\cdot x + 9\cdot 7$ $\square\\ 9(x + 7)$ $\square\\ 9(7x + 1)$ $\square\\ 72x$

(b) select all equivalent expressions from the list for the expression.
$14 + 10y - 8 - y$
$\square\\ 6y + 9y$ $\square\\ 9 + 6y$ $\square\\ 6y + 9$ $\square\\ 9y + 6$

Explanation:

Part (a)

Step1: Analyze \(9 \cdot x + 9 \cdot 7\)

Simplify \(9 \cdot x + 9 \cdot 7\), we know that \(9\cdot7 = 63\), so \(9 \cdot x + 9 \cdot 7=9x + 63\), which is equivalent to the original expression.

Step2: Analyze \(9(x + 7)\)

Using the distributive property \(a(b + c)=ab+ac\), here \(a = 9\), \(b=x\), \(c = 7\), so \(9(x + 7)=9x+9\times7=9x + 63\), which is equivalent to the original expression.

Step3: Analyze \(9(7x + 1)\)

Using the distributive property, \(9(7x + 1)=9\times7x+9\times1 = 63x+9
eq9x + 63\), so it is not equivalent.

Step4: Analyze \(72x\)

\(72x\) is a single - term expression with coefficient 72 and variable \(x\), while \(9x + 63\) is a two - term expression, so they are not equivalent.

First, simplify the expression \(14 + 10y-8 - y\).

Step1: Combine like terms for constants and like terms for variables.

For the constant terms: \(14-8=6\)
For the variable terms: \(10y-y = 9y\)
So \(14 + 10y-8 - y=9y + 6\)
Also, we can rewrite \(9y+6\) as \(6 + 9y\) or \(9y+6\) (addition is commutative, \(a + b=b + a\)).
Now let's analyze each option:

• For \(6y + 9y\): \(6y+9y = 15y

eq9y + 6\)

• For \(9 + 6y\): \(9+6y\) has a constant term 9 and variable term \(6y\), while our simplified expression has constant term 6 and variable term \(9y\), so it is not equivalent.
• For \(6y + 9\): \(6y + 9\) has variable term \(6y\) and constant term 9, not equivalent to \(9y+6\)
• For \(9y + 6\): It is the same as our simplified expression. Also, since addition is commutative, \(9y + 6=6 + 9y\) (but in the options, the correct one is \(9y + 6\) and also we can see that when we simplify \(14 + 10y-8 - y\), we get \(9y+6\))

Answer:

(a):
\(9\cdot x + 9\cdot 7\), \(9(x + 7)\)

Part (b)