Enter the numbers that make the algebraic expressions equivalent.(1 point)

Expression 1: 4x+5(x−2) +____

Expression 2:
____x−11−x+9

1 answer

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:

  1. \(B - 1 = 9\)
  2. \(A - 10 = -2\)

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