Question

Use the distributive property to prove that these algebraic expressions are equivalent explain your answer in one sentence for each expression and show your work expression 1 x - 5 * x + 3 expression 2 2x squared minus x squared minus 2X - 20 + 5

Answer 1

To prove that the two algebraic expressions are equivalent using the distributive property, we'll simplify both expressions step by step.

### Expression 1: \( x - 5(x + 3) \)

1. Distributing \(-5\) in the expression:
\[
x - 5(x + 3) = x - 5x - 15
\]
2. Combine like terms:
\[
x - 5x - 15 = -4x - 15
\]

### Expression 2: \( 2x^2 - x^2 - 2x - 20 + 5 \)

1. Combine \(2x^2\) and \(-x^2\):
\[
2x^2 - x^2 = x^2
\]
2. Now rewrite the expression:
\[
x^2 - 2x - 20 + 5
\]
3. Combine the constants \(-20\) and \(5\):
\[
-20 + 5 = -15
\]
4. Final expression:
\[
x^2 - 2x - 15
\]

Now, let's rewrite both simplified expressions:

- Expression 1 simplified is \(-4x - 15\).
- Expression 2 simplified is \(x^2 - 2x - 15\).

### Conclusion:
The expressions \(x - 5(x + 3)\) simplifies to \(-4x - 15\), and \(2x^2 - x^2 - 2x - 20 + 5\) simplifies to \(x^2 - 2x - 15\). Since there are different degrees (\(-4x - 15\) is linear while \(x^2 - 2x - 15\) is quadratic), these expressions are not equivalent. Thus, they represent distinct expressions in algebra.