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Chapter 1: Problem 53

Determine whether the given number is a solution of the equation. $$5 a-4=2 a+5 ; 3$$

Short Answer

Expert verified
Yes, the given number \(a = 3\) is a solution to the equation.

Step by step solution

01

Understand the equation

The equation is \(5a - 4 = 2a + 5\). You are supposed to determine if \(a = 3\) is a solution.
02

Substitute the given value

Substitute \(a = 3\) into the equation. This gives \(5(3) - 4 = 2(3) + 5\). Performing the operations inside the brackets will lead to the new equation, \(15 - 4 = 6 + 5\).
03

Simplify the equation

By simplifying the equation, \(15 - 4 = 11\) and \(6 + 5 = 11\). As you can see, both sides of the equation are equal, therefore, \(a = 3\) is a solution to the equation.

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Key Concepts

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

Algebraic Solutions
An algebraic solution involves solving equations or systems of equations to find the values of unknown variables. In the context of our exercise, determining if a given number is a solution to an equation falls under finding algebraic solutions. To test whether a particular value is a solution, you replace the variable with the given number and perform the necessary calculations. If both sides of the equation balance out, then you have found an algebraic solution. For instance, with the equation 5a - 4 = 2a + 5, when we substitute a = 3, and achieve equality on both sides after calculation, we can confidently say that 3 is an algebraic solution to the equation.
Equation Simplification
Equation simplification is reducing complex expressions to their simplest form, making them easier to solve or evaluate. It involves combining like terms, distributing variables, and simplifying expressions to isolate variables. In our exercise, we begin with 5(3) - 4 = 2(3) + 5, which simplifies to 15 - 4 = 6 + 5. This process often involves basic arithmetic such as addition, subtraction, and sometimes factoring. The goal is to make both sides of the equation as straightforward as possible. Once simplified, it becomes clear whether the value chosen for the variable satisfies the equation.
Substitution Method
The substitution method is a technique for solving algebraic equations by replacing variables with their known or assumed values. It is particularly useful for determining if a given number is a solution to the equation. In the example we have, substituting a = 3 into the original equation 5a - 4 on the left and 2a + 5 on the right leads to a new expression, allowing for a direct comparison. If your assumptions are correct and the resulting values on both sides of the equation match after the substitution, then the value assumed for the variable is indeed a solution.
Verifying Solutions of Equations
Verifying solutions includes checking each step of an equation to confirm that the chosen variable value is correct. After substituting and simplifying, you compare the left and right sides of the equation to see if they are equal. If they match, the substituted value is a verified solution. In the problem 5a - 4 = 2a + 5, once we reached 11 = 11 after substituting a = 3, we've verified that 3 is indeed a solution. This verification step is crucial in ensuring the integrity of the solution and should always be performed as a part of the equation-solving process.

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