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Chapter 7: Problem 48

Solve the equation. $$ 15-2 a=7 $$

Short Answer

Expert verified
The solution to the equation \(15 - 2a = 7\) is \(a = 4\).

Step by step solution

01

Isolation

Start by isolating the variable on one side. Subtract 15 from both sides of the equation to move the 15 from the left side of the equation to the right. This will give: \(-2a = 7 - 15\).
02

Simplification

Next, simplify the right side of the equation by subtracting 15 from 7. This results in: \(-2a = -8\).
03

Final Calculation

Finally, solve for \(a\) by dividing the equation by -2 on both sides. This will give: \(a = -8 / -2\). So, \(a = 4\).

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Key Concepts

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

Isolation of Variable
In solving linear equations, isolating the variable is often the first crucial step. Here, our goal is to get the variable, like "\(a\)" in this case, by itself on one side of the equation. To do this:
  • Identify the variable you need to isolate.
  • Move any terms that do not involve this variable to the other side of the equation using basic arithmetic operations like addition or subtraction.
In our exercise, we start with the equation \(15 - 2a = 7\). By subtracting 15 from both sides, we effectively "move" the 15 to the other side by keeping the equation balanced. This step transforms the equation into \(-2a = 7 - 15\), bringing us a step closer to isolating "\(a\)".
Simplification of Expressions
Once the variable part is isolated, the next step is to simplify the expressions you have created. Simplification involves combining like terms and performing arithmetic operations to make the equation easier to work with.
  • Look at each side of the equation independently.
  • Perform operations such as addition, subtraction, multiplication, and division.
  • Combine terms whenever possible.
In our example, we have \(-2a = 7 - 15\). Simplifying the right side, we subtract directly: \(7 - 15\) which equals \(-8\). Thus, the equation is simplified to \(-2a = -8\). This simple arithmetic operation helps in reducing the complexity of the equation, paving the way to find the value of \(a\).
Equation Balancing
Equation balancing is about keeping both sides of an equation equal while you solve for the unknown. No matter what operation is applied to one side, it should also be applied to the other side to maintain balance. This principle ensures the equality of the equation isn't disrupted.
  • Whatever you do to one side of the equation, do to the other.
  • Apply inverse operations to simplify and solve the equation.
  • Think of the equation as a scale where both sides need to be equal.
In the exercise, after simplifying, we have: \(-2a = -8\). To solve for \(a\), divide both sides by \(-2\). This means we're applying the same operation to both sides to solve for the variable without affecting the balance. So, \(a = \frac{-8}{-2}\), resulting in \(a = 4\). Maintaining this balance is key in solving and verifying the accuracy of your equations.

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