To make the two algebraic expressions equivalent, we can simplify both expressions step by step.
Expression 1:
\[ 4x + 5(x - 2) + ___ \]
First, let's simplify \(5(x - 2)\):
\[ 5(x - 2) = 5x - 10 \]
Now, substitute this back into Expression 1:
\[ 4x + 5x - 10 + ___ \]
Combine like terms:
\[ (4x + 5x) - 10 + __ = 9x - 10 + __ \]
Let's denote the blank with \(A\):
\[ 9x - 10 + A \]
Expression 2:
\[ __x - 11 - x + 9 \]
Let's denote the blank with \(B\):
\[ Bx - 11 - x + 9 \]
Now combine like terms:
\[ (B - 1)x + (-11 + 9) = (B - 1)x - 2 \]
Now, we want the two expressions to be equivalent:
From Expression 1: \[ 9x - 10 + A \Rightarrow 9x + (A - 10) \]
From Expression 2: \[ (B - 1)x - 2 \Rightarrow (B - 1)x + (-2) \]
To make the \(x\) coefficients equal and the constant terms equal, we set up the following equations:
- \(B - 1 = 9\)
- \(A - 10 = -2\)
Solve for \(B\):
\[ B - 1 = 9 \implies B = 9 + 1 = 10 \]
Solve for \(A\):
\[ A - 10 = -2 \implies A = -2 + 10 = 8 \]
Therefore:
- The number for Expression 1 (blank) is 8.
- The number for Expression 2 (blank) is 10.
The completed expressions are:
- Expression 1: \( 4x + 5(x - 2) + 8 \)
- Expression 2: \( 10x - 11 - x + 9 \)
So the final answers are:
- 8 for the first blank
- 10 for the second blank