Answered

Which statement is true for the equations given below?

$\begin{array}{l}

3 x+2=-6 \\

3 x+6=-2

\end{array}$


ABoth equations have the same solution because the second equation can be obtained by dividing both sides of the first equation by 3.
BBoth equations have the same solution because the second equation can be obtained by adding 4 to both sides of the first equation.
CBoth equations do not have the same solution because the second equation can be obtained by adding 4 to both sides of the first equation.
DBoth equations have the same solution because the second equation can be obtained by multiplying both sides of the first equation by 3.

Verified Answer :

  • Solve the first equation \(3x + 2 = -6\) to find \(x = -8/3\).
  • Solve the second equation \(3x + 6 = -2\) to find \(x = -8/3\).
  • Recognize that adding \(4\) to both sides of the first equation, \(3x+2+4 = -6+4\), transforms it into the second equation, \(3x+6 = -2\).
  • Conclude that both equations have the same solution because the second equation can be obtained by adding 4 to both sides of the first equation: \(\boxed{\text{Both equations have the same solution because the second equation can be obtained by adding 4 to both sides of the first equation.}}\)

Explanation

  1. Solving the Equations

We are given two linear equations:

  1. \(3x + 2 = -6\)
  2. \(3x + 6 = -2\)

To solve Equation 1, we isolate the variable \(x\):
\[3x + 2 = -6\]
Subtract 2 from both sides:
\[3x = -6 - 2 = -8\]
Divide by 3:
\[x = -\frac{8}{3}\]

Now, let's solve Equation 2, following the same process:
\[3x + 6 = -2\]
Subtract 6 from both sides:
\[3x = -2 - 6 = -8\]
Divide by 3:
\[x = -\frac{8}{3}\]

We see that both equations yield the same solution, \(x = -\frac{8}{3}\).

  1. Comparing the Equations

We examine how the first equation relates to the second equation. The first equation is \(3x + 2 = -6\). If we add 4 to both sides of this equation, we get:
\[(3x + 2) + 4 = -6 + 4\]
\[3x + 6 = -2\]
This is exactly the second equation. Since adding the same constant to both sides of an equation does not change its solution, this confirms why both equations possess identical solutions.

  1. Concluding the True Statement

Given our findings that both equations have the same solution (\(x = -8/3\)) and that the second equation is formed by adding 4 to both sides of the first, we can identify the correct statement from the provided options. The correct statement is: "Both equations have the same solution because the second equation can be obtained by adding 4 to both sides of the first equation."

Examples

This problem demonstrates how simple algebraic transformations preserve the equality of an equation, a concept essential for solving systems of equations. In real life, this is like balancing a scale: if you add the same amount of weight to both sides, the scale remains balanced at the same point. This ensures that the 'balance' of the equation—its solution—remains consistent throughout your manipulations.

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