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Chapter 2: Problem 57

Solve each equation. Check all solutions. SEE EXAMPLE 4. (OB]ECTIVE 2) $$3(a+2)=2(a-7)$$

Short Answer

Expert verified
Based on the detailed solution provided, the solution to the equation \(3(a+2)=2(a-7)\) is \(a=-20\).

Step by step solution

01

Expand the brackets

Expand both sides of the equation which involves multiplying 3 by \(a\) and 2 (on the left side), and 2 by \(a\) and -7 (on the right side) respectively. This simplifies the equation to \(3a+6=2a-14\).
02

Collect like terms

Move \(2a\) from the right side of the equation to the left side by subtracting \(2a\) from both sides, and move 6 from the left side to the right side by subtracting 6 from both sides. This simplifies the equation further into \(3a-2a=-14-6\). Further simplification results in \(a=-20\).
03

Check the solution

The last step is to verify the solution by substitifying \(a=-20\) into original equation \(3(a+2)=2(a-7)\). If the left and right sides are equal when \(a=-20\), then the solution is correct.

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Key Concepts

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

Understanding Algebra
Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. These symbols typically stand for variables, which can change.
In the provided exercise, we are required to solve a linear algebraic equation.
The goal is to find the value of the variable that makes the equation true. Linear equations like this one are straightforward as they form a straight line when graphed, indicating a constant rate of change.
By familiarizing oneself with algebra, solving such equations becomes less of a challenge and more of a systematic method of isolating the variable.
Simplifying Equations
Equation simplification is a valuable skill in algebra, allowing for more manageable computations.
The initial step in our exercise involves expanding brackets, which is vital to simplification.
When faced with an equation like \(3(a+2)=2(a-7)\), the brackets are expanded by distributing the coefficients:
  • Multiply 3 by both \(a\) and 2.
  • Multiply 2 by \(a\) and -7.
This transforms the equation into \(3a+6=2a-14\).
With the brackets out of the way, we then combine like terms. This step is crucial, as it reduces the equation to a form where the same types of terms are together. This process eases the path for variable isolation, a core practice in simplifying equations.
Variable Manipulation Techniques
Variable manipulation is an integral aspect of solving equations. The main objective is to isolate the variable on one side of the equation to determine its value.
In our example problem, after expanding and simplifying, the equation becomes \(3a+6=2a-14\).
To isolate \(a\), we take these steps:
  • Subtract \(2a\) from both sides to gather all \(a\) terms on one side, resulting in \(3a-2a\).
  • Subtract 6 from each side to move constant terms away from \(a\), simplifying it further to \(a=-20\).

By methodically performing these operations, we achieve the final answer for \(a\). It is important to verify the solution by substituting back into the original equation, ensuring equality on both sides and confirming the answer's accuracy.

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