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Chapter 3: Problem 138

Is \(-7\) a solution of the equation \(5+a=-9-a ?\)

Short Answer

Expert verified
Yes, -7 is a solution for the equation.

Step by step solution

01

Substitute the provided value into the equation

Replace 'a' in the equation with \(-7\) to get: \(5 + (-7) = -9 - (-7)\).
02

Simplify the equation

By performing the addition and subtraction operations in the equation, it simplifies to \(-2 = -9 +7\).
03

Simplify further to check the equality

When the right side of the equation is simplified, it becomes: \(-2 = -2\).
04

Concluding the result

As the left side of the equation equals the right side, the provided value, \(-7\), is indeed a solution for the equation.

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Key Concepts

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

Substitution Method
The substitution method is a fundamental algebraic technique used to determine if a certain value is a solution to an equation. By substituting the alleged solution directly into the equation, we can assess the equation's validity.

For instance, if you're given an equation such as \(5 + a = -9 - a\) and the number \(-7\) to test, you would start by replacing every instance of 'a' with \(-7\). The paramount step here is accurate substitution ensuring each 'a' is replaced with \(-7\), which changes the equation to \(5 + (-7) = -9 - (-7)\). This must be done carefully because any mistake at this stage can lead to an incorrect evaluation of the solution's validity.
Simplifying Equations
After substituting values, simplifying the equation is crucial for solving algebra problems. Simplification involves combining like terms and performing basic arithmetic operations: addition, subtraction, multiplication, and division.

In the example \(5 + (-7) = -9 - (-7)\), we simplify each side of the equation separately. The left side becomes \(5 - 7\), which simplifies to \(-2\). The right side involves the subtraction of a negative number, which is equivalent to addition, so \(-9 + 7\), also simplifying to \(-2\).

Simplification not only makes the equation easier to read but also facilitates the final steps of solving or checking the solution.
Checking Solutions
Once the equation is simplified, double-checking the solution is a good practice to confirm its correctness. This involves verifying if the simplified left-hand side (LHS) and right-hand side (RHS) of the equation are equal.

In the given exercise, after simplification, we have \(-2 = -2\). The LHS equals the RHS, indicating that the value we substituted for 'a' does make the original equation true. Therefore, \(-7\) is indeed a solution to the equation \(5 + a = -9 - a\). It is vital to remember that the equality must hold true for a value to be considered a solution鈥攖his is the essence of checking solutions in algebra.

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