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Paula used these steps to solve an equation:

Step 1: $-4(x+8)-2 x=25$

Step 2: $-4 x-32-2 x=25$

Step 3: $-6 x-32=25$

Step 4: $-6 x=57$

Step 5: $x=-9 \frac{1}{2}$



Between which two steps did Paula use the division property of equality?


Asteps 1 and 2
Bsteps 2 and 3
Csteps 3 and 4
Dsteps 4 and 5

Verified Answer :

  • Identify the algebraic property applied at each transition of the equation-solving process.
  • Observe that the transition from Step 4 to Step 5 involves dividing both sides of the equation by the coefficient \(-6\).
  • Conclude that the division property of equality is applied between steps 4 and 5.
  • Final answer: \(\boxed{\text{steps 4 and 5}}\).

Explanation

  1. Analyzing the Problem and Properties

To determine where the division property of equality was applied, we analyze the transition between each consecutive step in the process of solving the equation \(-4(x+8)-2x=25\).
The division property of equality states that if \(a = b\) and \(c eq 0\), then \(\frac{a}{c} = \frac{b}{c}\). This property allows us to isolate a variable by dividing both sides of an equation by the same non-zero number.

  1. Evaluating Step Transitions

We examine each step transition:

  • Steps 1 and 2: The equation transitions from \(-4(x+8)-2x=25\) to \(-4x-32-2x=25\). This shows the distribution of \(-4\) over \((x+8)\), which is an application of the distributive property, not the division property.
  • Steps 2 and 3: The equation transitions from \(-4x-32-2x=25\) to \(-6x-32=25\). This shows the combination of like terms (\(-4x\) and \(-2x\) are added to get \(-6x\)), not division.
  • Steps 3 and 4: The equation transitions from \(-6x-32=25\) to \(-6x=57\). This shows the addition of 32 to both sides of the equation (\(-32 + 32 = 0\) and \(25 + 32 = 57\)), which is an application of the addition property of equality.
  • Steps 4 and 5: The equation transitions from \(-6x=57\) to \(x=-9 \frac{1}{2}\). Here, the coefficient of \(x\), which is \(-6\), is removed by dividing both sides by \(-6\). This is \(\frac{-6x}{-6} = \frac{57}{-6}\), resulting in \(x = -9 \frac{1}{2}\). This is a direct application of the division property of equality.
  1. Conclusion

By examining the transitions, we clearly see that the division property of equality was applied specifically to move from Step 4 to Step 5. Therefore, the correct choice is steps 4 and 5. The final answer is \(\boxed{\text{steps 4 and 5}}\).

Examples

Understanding the properties of equality is essential when working with financial budgets. For example, if you have a total monthly budget of \(\$1200\) and you want to calculate how much you can spend per day over 30 days, you set up the equation \(30x = 1200\). To solve for \(x\), you apply the division property of equality by dividing both sides by 30, giving \(x = \frac{1200}{30} = 40\). Mastering these algebraic steps ensures your planning and resource management remain accurate and balanced.