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Chapter 2: Problem 75

Solve each equation, and check the solution. $$ 0.9 w-0.5 w+0.1 w=-3 $$

Short Answer

Expert verified
The solution to the equation is w = -6, and it is verified.

Step by step solution

01

- Combine like terms

First, combine the like terms on the left side of the equation. The terms are 0.9w, -0.5w, and 0.1w.Calculate:a = 0.9 - 0.5 + 0.1 = 0.5
02

- Form the simplified equation

Rewrite the equation using the combined terms:a*w = 0.5w = -3
03

- Solve for w

Isolate the variable w by dividing both sides of the equation by the coefficient (0.5):a*w / a = (-3) / 0.5This simplifies to: w = (-3) / (0.5) = (-6)
04

- Checking the solution

Substitute w = -6 back into the original equation:0.9*(-6) - 0.5*(-6) + 0.1*(-6) = -3Simplify each term:-5.4 + 3 - 0.6 = -3Combine the results:-3 = -3Since both sides of the equation are equal, the solution is verified.

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Key Concepts

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

Combining Like Terms
When solving algebraic equations, it's essential to simplify the expression by combining like terms. Like terms are those that have the same variable raised to the same power. In the equation \(0.9w - 0.5w + 0.1w = -3\), the terms 0.9w, -0.5w, and 0.1w are all like terms because they each include the variable \(w\). By adding or subtracting their coefficients, you can simplify the equation:
  • Combine the coefficients: \(0.9 - 0.5 + 0.1 = 0.5\)
  • This gives a single term: \(0.5w\)

So, the simplified equation is \(0.5w = - 3\). Combining like terms reduces the complexity of the equation, making it easier to solve for the variable.
Isolating the Variable
The next step in solving the equation \(0.5 w = -3\) is to isolate the variable \(w\). This means you want to get \(w\) by itself on one side of the equation. Since \(w\) is multiplied by 0.5, you need to divide both sides of the equation by 0.5:
  • Left side: \(\frac{0.5w}{0.5} = w\)
  • Right side: \(\frac{-3}{0.5} = -6\)

So, \(w = -6\). By isolating the variable \(w\), you have found the solution to the equation. It's important to perform the same operation on both sides of the equation to maintain equality.
Verifying Solutions
Once you've found the value of the variable, the final step is to verify the solution. This means substituting the value back into the original equation to check if both sides are equal. For our example:
Substitute \(w = -6\) into the original equation:
  • Calculate \(0.9(-6) - 0.5(-6) + 0.1(-6)\)
  • Simplify each term: \(-5.4 + 3 - 0.6\)
  • This equals -3, which matches the right side of the original equation.

Since both sides are equal, the solution \(w = -6\) is verified as correct. Always verify your solution to ensure accuracy and to confirm that no mistakes were made during the calculation process.

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Most popular questions from this chapter

Beyonc茅 and Guns N' Roses had the two top-grossing concert tours for 2016 , together generating \(\$ 300\) million in ticket sales. Guns \(\mathrm{N}\) ' Roses took in \(\$ 38\) million less than Beyonc茅. How much did each tour generate? (Data from Pollstar.)

Solve each formula for the specified variable. \(S=\frac{1}{2}(a+b+c)\) for \(b\)

Beyonc茅 and Guns \(\mathrm{N}\) ' Roses had the two top-grossing concert tours for 2016 , together generating \(\$ 300\) million in ticket sales. Guns \(\mathrm{N}\) ' Roses took in \(\$ 38\) million less than Beyonc茅. How much did each tour generate? (Data from Pollstar.)

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