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Chapter 3: Problem 25

Solve and check each equation. $$-5 z=35$$

Short Answer

Expert verified
The solution to the equation \(-5z = 35\) is \(z = -7\).

Step by step solution

01

Write down the given equation

The given equation is: \(-5z = 35\)
02

Solve for z

To isolate z, we will divide both sides of the equation by -5: \[\frac{-5z}{-5} = \frac{35}{-5}\] This simplifies to: \[z = -7\] So, the value of z is -7.
03

Check the solution

Now, we will check our solution by substituting -7 for z in the given equation and see if it holds true: \[-5(-7) = 35\] Multiplying -5 by -7, we get 35: \[35 = 35\] The equation is true, so our solution is correct. The value of z is -7.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Checking Solutions
After finding a solution to an equation, it's crucial to double-check it. This helps ensure that the answer is accurate. Consider the equation we worked on:
  • First, identify the solution for the variable, in this case, \(-7\) for \(z\).
  • Next, substitute the solution back into the original equation, \(-5z = 35\).
  • If substituting \(z = -7\) results in both sides of the equation being equal, then your solution is correct.
Checking your solution provides confidence that no mistakes were made during calculation. It is as simple as replacing and recalculating. Always aim to verify, as errors can occur easily.
Division in Algebra
Solving many algebraic equations involves using division as a key step. It works great for simplifying an equation. Let鈥檚 see how division helps to work with equations like \(-5z = 35\):
  • To isolate the variable, divide both sides of the equation by the coefficient of the variable which, in this case, is \(-5\).
  • The division looks like \(\frac{-5z}{-5} = \frac{35}{-5}\).
  • This step simplifies to \(z = -7\), effectively solving for \(z\).
Through division, solutions become clearer and easier to spot. Ensure both sides of the equation get the same treatment to maintain balance and accuracy.
Isolating Variables
To find the value of the variable in a linear equation, isolating it is key. The process involves moving terms appropriately so that the variable stands alone on one side of the equation. Here鈥檚 how it's done:
  • Identify the term containing the variable, like \(-5z\) in \(-5z = 35\).
  • Apply inverse operations to both sides to move other terms. In this equation, division by \(-5\) helps isolate \(z\).
  • Aim for a result where the variable is equal to a constant, eventually giving \(z = -7\).
Isolating the variable simplifies the equation and brings clarity to its solution. This technique is fundamental in solving algebraic problems, keeping you on the clear path to finding answers.

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