Answered

Match each expression to its simplified version:



Expressions to match:

  1. $4(2 x-1)$
  2. $3(2 x-5)$
  3. $2(x+1)$
  4. $-6(2 x-3)$



Simplified versions:

  • $8 x - 4$
  • $-12 x + 18$
  • $6 x - 15$
  • $2 x + 2$

Verified Answer :

  • Distribute the factors in each given expression: \(4(2x - 1) = 8x - 4\), \(3(2x - 5) = 6x - 15\), \(2(x + 1) = 2x + 2\), and \(-6(2x - 3) = -12x + 18\).
  • Match the expanded results to the original linear expressions.
  • The pairings are: \(8x - 4 \leftrightarrow 4(2x - 1)\); \(-12x + 18 \leftrightarrow -6(2x - 3)\); \(6x - 15 \leftrightarrow 3(2x - 5)\); \(2x + 2 \leftrightarrow 2(x + 1)\).
  • \(\boxed{8x - 4=4(2x - 1), -12x + 18=-6(2x - 3), 6x - 15=3(2x - 5), 2x + 2=2(x + 1)}\)

Explanation

  1. Analysis of the expressions

We are given four linear expressions: \(8x - 4\), \(-12x + 18\), \(6x - 15\), and \(2x + 2\). We are also given four factored expressions: \(4(2x - 1)\), \(3(2x - 5)\), \(2(x + 1)\), and \(-6(2x - 3)\). Our objective is to match each linear expression to its equivalent factored form using the distributive property, which states that \(a(bx + c) = abx + ac\).

  1. Expanding the factored expressions

To find the matches, we expand each of the four factored expressions by distributing the coefficient into the binomial:

  1. \(4(2x - 1) = 4(2x) + 4(-1) = 8x - 4\)
  2. \(3(2x - 5) = 3(2x) + 3(-5) = 6x - 15\)
  3. \(2(x + 1) = 2(x) + 2(1) = 2x + 2\)
  4. \(-6(2x - 3) = -6(2x) + (-6)(-3) = -12x + 18\)
  1. Matching the expressions

Now, we compare the expanded results with the original linear expressions:

  • The expression \(8x - 4\) corresponds to \(4(2x - 1)\).
  • The expression \(-12x + 18\) corresponds to \(-6(2x - 3)\).
  • The expression \(6x - 15\) corresponds to \(3(2x - 5)\).
  • The expression \(2x + 2\) corresponds to \(2(x + 1)\).

Examples

In daily life, factoring expressions is like simplifying a shopping list. If you are buying 4 packs that each contain 2 apples and 1 orange, you can express the total as 4(2a + 1o), which is the same as 8a + 4o. Understanding how to expand and factor these terms helps in quickly calculating quantities and organizing items efficiently, just as algebra helps us structure mathematical information.

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