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Chapter 3: Problem 9

Determine if the given value is a solution to the equation. $$a-4=-9 ; a=5$$

Short Answer

Expert verified
No, the given value \(a = 5\) is not a solution to the equation \(a - 4 = -9\), because after substituting, we get \(1 = -9\), which is not true.

Step by step solution

01

Identify the equation and the given value

We are given the equation: \(a - 4 = -9\) and the value of \(a = 5\).
02

Substitute the given value into the equation

Now we will substitute the given value of \(a = 5\) into the equation: \((5) - 4 = -9\).
03

Simplify and verify equality

After substituting, our equation becomes: \[1 = -9\]. Since both sides of the equation are not equal, the given value of \(a = 5\) is not a solution to the equation \(a - 4 = -9\).

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Key Concepts

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

Solution Verification
When checking if a number is a solution to an equation, we need to ensure both sides of the equation are equal after substituting the given number. This process is known as solution verification. To verify, you start by substituting the provided value into the equation and then simplify it. If both sides of the equation match, the number is a correct solution. If they do not match, as in our original exercise with the equation \( a - 4 = -9 \) and \( a = 5 \), the number is not a solution. Key points to consider during solution verification:
  • Always substitute the number in place of the variable.
  • Simplify the equation fully to compare both sides clearly.
  • If actions on both sides don鈥檛 yield equal results, the value isn鈥檛 a solution.
Substitution in Equations
Substitution plays a crucial role in solving equations or verifying potential solutions. It involves replacing the variable in the equation with a specified value. This process is straightforward, but it's important to follow it carefully to avoid errors. In the problem where we have the equation \( a - 4 = -9 \), substituting \( a = 5 \) gives us \( 5 - 4 \). This step shows how the equation changes and allows us to proceed with simplification.Important aspects of substitution include:
  • Making sure the correct value is substituted in place of the variable.
  • Replacing every instance of the variable if it occurs multiple times in the equation.
  • Ensuring neatness and accuracy to avoid miscalculations when simplifying later.
Simplification of Equations
Simplification is the process of reducing an equation to its simplest form. This is essential in determining the solution of an equation or verifying a potential solution. During simplification, mathematical operations like addition, subtraction, multiplication, or division are performed as needed.In our exercise, after substituting \( a = 5 \) into \( a - 4 = -9 \), we simplified it to \( 1 = -9 \). This simplification clearly shows a discrepancy, highlighting why \( a = 5 \) is not a solution.Common steps involved in simplification include:
  • Carrying out arithmetic operations (e.g., subtracting or adding similar terms).
  • Combining like terms for a cleaner and clearer equation.
  • Clearly identifying whether the two sides of the equation match after simplification.
Simplification forms the backbone of many algebraic processes, helping to clarify and solve equations efficiently.

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