Question

name: ____ per: ____
date: ______
directions: identify the variable terms, constant terms, and coefficients for each expression.

expression	variable terms	coefficients	constant terms
$8v + 7 - 2v$
$-6n - 25 + 7n - 4$
$3a - 4b + 17 + a - 1$
$1c - 4d + c - 9d - 2c$
$-5t - 3 + 2y - y + 8x - 14$

directions: simplify each expression by combining like terms.
例 $-6c + 4c$

Answer:

To solve this, we'll analyze each expression to identify variable terms, coefficients, and constant terms, then simplify by combining like terms. Let's go through each expression one by one.

Expression 1: \( 4x + 15x \)

Step 1: Identify Variable Terms, Coefficients, Constant Terms

• Variable Terms: \( 4x \), \( 15x \) (both have the variable \( x \))
• Coefficients: 4 (coefficient of \( 4x \)), 15 (coefficient of \( 15x \))
• Constant Terms: None (no terms without a variable)

Step 2: Combine Like Terms

Like terms have the same variable (or no variable). Here, \( 4x \) and \( 15x \) are like terms. Add their coefficients:
\( 4x + 15x = (4 + 15)x = 19x \)

Expression 2: \( 8v + 7 - 2v \)

Step 1: Identify Variable Terms, Coefficients, Constant Terms

• Variable Terms: \( 8v \), \( -2v \) (variable \( v \))
• Coefficients: 8 (coefficient of \( 8v \)), \( -2 \) (coefficient of \( -2v \))
• Constant Terms: \( 7 \) (term without a variable)

Step 2: Combine Like Terms

Combine the variable terms first:
\( 8v - 2v = (8 - 2)v = 6v \)
Then, the constant term remains \( 7 \). So the simplified expression is \( 6v + 7 \).

Expression 3: \( -6n - 25 + 7n - 4 \)

Step 1: Identify Variable Terms, Coefficients, Constant Terms

• Variable Terms: \( -6n \), \( 7n \) (variable \( n \))
• Coefficients: \( -6 \) (coefficient of \( -6n \)), \( 7 \) (coefficient of \( 7n \))
• Constant Terms: \( -25 \), \( -4 \) (terms without a variable)

Step 2: Combine Like Terms

Combine variable terms:
\( -6n + 7n = (-6 + 7)n = n \)
Combine constant terms:
\( -25 - 4 = -29 \)
So the simplified expression is \( n - 29 \).

Expression 4: \( 3a - 4b + 17 + a - 1 \)

Step 1: Identify Variable Terms, Coefficients, Constant Terms

• Variable Terms: \( 3a \), \( a \) (variable \( a \)); \( -4b \) (variable \( b \))
• Coefficients: 3 (coefficient of \( 3a \)), 1 (coefficient of \( a \)); \( -4 \) (coefficient of \( -4b \))
• Constant Terms: \( 17 \), \( -1 \) (terms without a variable)

Step 2: Combine Like Terms

Combine \( a \)-terms:
\( 3a + a = (3 + 1)a = 4a \)
The \( -4b \) term has no like terms (no other \( b \)-terms), so it remains.
Combine constant terms:
\( 17 - 1 = 16 \)
So the simplified expression is \( 4a - 4b + 16 \).

Expression 5: \( 1c - 4d + c - 9d - 2c \) (Note: \( 1c = c \))

Step 1: Identify Variable Terms, Coefficients, Constant Terms

• Variable Terms: \( c \), \( c \), \( -2c \) (variable \( c \)); \( -4d \), \( -9d \) (variable \( d \))
• Coefficients: 1 (coefficient of \( c \)), 1 (coefficient of \( c \)), \( -2 \) (coefficient of \( -2c \)); \( -4 \) (coefficient of \( -4d \)), \( -9 \) (coefficient of \( -9d \))
• Constant Terms: None (no terms without a variable)

Step 2: Combine Like Terms

Combine \( c \)-terms:
\( c + c - 2c = (1 + 1 - 2)c = 0c = 0 \) (the \( c \)-terms cancel out)
Combine \( d \)-terms:
\( -4d - 9d = (-4 - 9)d = -13d \)
So the simplified expression is \( -13d \) (since the \( c \)-terms sum to 0).

Expression 6: \( -5t + 3 + 2y - y + 8x - 14 \)

Step 1: Identify Variable Terms, Coefficients, Constant Terms

• Variable Terms: \( -5t \) (variable \( t \)); \( 2y \), \( -y \) (variable \( y \)); \( 8x \) (variable \( x \))
• Coefficients: \( -5 \) (coefficient of \( -5t \)); 2 (coefficient of \( 2y \)), \( -1 \) (coefficient of \( -y \)); 8 (coefficient of \( 8x \))
• Constant Terms: \( 3 \), \( -14 \) (terms without a variable)

Step 2: Combine Like Terms

Combine \( y \)-terms:
\( 2y - y = (2 - 1)y = y \)
Combine constant terms:
\( 3 - 14 = -11 \)
The \( -5t \) and \( 8x \) terms have no like terms, so they remain.
So the simplified expression is \( -5t + 8x + y - 11 \).

Extra: \( -6c + 4c \) (from the bottom right)

Step 1: Identify Variable Terms, Coefficients, Constant Terms

• Variable Terms: \( -6c \), \( 4c \) (variable \( c \))
• Coefficients: \( -6 \) (coefficient of \( -6c \)), \( 4 \) (coefficient of \( 4c \))
• Constant Terms: None

Step 2: Combine Like Terms

Add the coefficients:
\( -6c + 4c = (-6 + 4)c = -2c \)

Summary of Simplified Expressions:

1. \( 4x + 15x = \boldsymbol{19x} \)
2. \( 8v + 7 - 2v = \boldsymbol{6v + 7} \)
3. \( -6n - 25 + 7n - 4 = \boldsymbol{n - 29} \)
4. \( 3a - 4b + 17 + a - 1 = \boldsymbol{4a - 4b + 16} \)
5. \( 1c - 4d + c - 9d - 2c = \boldsymbol{-13d} \)
6. \( -5t + 3 + 2y - y + 8x - 14 = \boldsymbol{8x - 5t + y - 11} \) (rewritten for clarity, order of terms doesn’t matter)
7. \( -6c + 4c = \boldsymbol{-2c} \)

If you need help with a specific part (e.g., identifying terms or simplifying a particular expression), let me know!