Use properties to rewrite the given equation. which equations have the same solution as x + + x = – x plus startfraction 2 over 3 endfraction plus x equals startfraction one-half endfraction minus startfraction 1 over 5 endfraction x. x? select three options.

The missing options are found in;

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This is based on algebraic properties of equality.

Options A, B & D are correct.

1) The equation we want to rewrite is;

³/₅x + ²/₃ + x = ¹/₂ – ¹/₅x

Collecting like terms we have;

(³/₅ + 1)x + ²/₃ = ¹/₂ – ¹/₅x

Simplifying the bracket gives;

⁸/₅x + ²/₃ = ¹/₂ – ¹/₅x

This corresponds with the solution in Option A

2) ³/₅x + ²/₃ + x = ¹/₂ – ¹/₅x

Let us use multiplication property of equality which states that if we multiply both sides of an equation by the same quantity, then both sides remain equal.

Let's get rid of the denominators by multiplying each term by the LCM of the denominators. LCM of 2,3,5 is 30. Thus;

30(³/₅x) + 30(²/₃) + 30x = 30(¹/₂) - 30(¹/₅x)

Simplifying gives;

18x + 20 + 30x = 15 - 6x

This solution corresponds with that in option B

3) We will make use of additive property of equality which states that when we add the same number to both sides of an equation, then both sides remain equal.

Let's add 6x to both sides of 18x + 20 + 30x = 15 - 6x;

18x + 20 + 30x + 6x = 15 - 6x + 6x

⇒ 24x + 30 + 20 = 15

Using subtractive property of equality, let us subtract 20 from both sides to get;

24x + 30 + 20 - 20 = 15 - 20

This corresponds with the solution in Option D.

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The answer is A  B and D

A. 8/5x + 2/3 = 1/2 – 1/5x

B. 18x + 20 + 30x = 15 – 6x

C. 18x + 20 + x = 15 – 6x

D. 24x + 30x = –5

E. 12x + 30x = –5

Step-by-step explanation:

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