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Chapter 3: Problem 2

Solve each equation. Check your solution. $$-2 a=10$$

Short Answer

Expert verified
The solution is \(a = -5\).

Step by step solution

01

Isolate the Variable

To solve the equation \[-2a = 10,\]we need to isolate the variable \(a\). To do this, we will divide both sides of the equation by -2, the coefficient of \(a\):\[a = \frac{10}{-2}\]Simplifying the right side gives:\[a = -5.\]
02

Verify the Solution

To ensure our solution is correct, we substitute \(a = -5\) back into the original equation:\[-2(-5) = 10.\]Calculating the left side gives:\[10 = 10,\]which confirms that our solution is indeed correct.

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Key Concepts

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

Isolating the Variable
Solving any equation primarily begins with isolating the variable. In our example of \[-2a = 10,\]the term involving the variable (here, \(-2a\)) needs to be isolated. Isolating simply means getting the variable on one side of the equation by itself while keeping the equation balanced. Here's how you can approach it:
  • Identify the coefficient of the variable, which in this case is \(-2\).
  • To remove this coefficient and isolate the variable, perform the inverse operation. Since the coefficient is a multiplication, you would use division.
Divide both sides of the equation by the coefficient:\[a = \frac{10}{-2}\]You'll calculate the division, leading to:\[a = -5.\]This process of isolating ensures that the equation resolves neatly into a solution that's easy to check.
Verifying the Solution
Once you have isolated the variable and found that \(a = -5\), it鈥檚 crucial to verify that this solution is correct. Verification acts as your safeguard against errors. You can do this verification by substituting the solution back into the original equation.
  • Start by replacing every instance of \(a\) in the original equation with \(-5\).
  • The equation transforms from \(-2a = 10\) to \(-2(-5) = 10\).
Next, calculate the left side:\[-2 \times -5 = 10\]Observe that it matches perfectly with the right side of the equation:\[10 = 10.\]Verifying the solution ensures it satisfies the original equation, providing a solid conclusion that the solution is correct.
Prealgebra Concepts
Grasping prealgebra concepts lays the foundation for future mathematical learning. Understanding such concepts will significantly help in developing problem-solving skills.
  • Inverse Operations: Knowing that operations have opposites (addition vs. subtraction, multiplication vs. division) is crucial in manipulating equations.
  • Balancing Equations: A fundamental idea in prealgebra is to treat both sides of an equation equally. What is done to one side must be done to the other.
  • Simplicity in Steps: Breaking problems into smaller, more manageable pieces, like isolating the variable before verifying a solution, makes solving equations more accessible.
Master these prealgebra concepts and you'll find that tackling more complex equations becomes less daunting. Such skills not only help in mathematics but also improve logical and critical thinking across various tasks and challenges.

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