Problem 92 - Introductory and Intermediate Algebra for College Students 4th

Question

Exercises \(91-93\) will help you prepare for the material covered in the next section. Is 6 a solution of \(2(x-3)-17=13-3(x+2) ?\)

Step-by-Step Solution

Answer

Yes, 6 is a solution to the given equation.

Step 1: Substitute x with 6

Plug in the value of x as 6 in the given equation. We get: \(2(6-3) - 17 = 13 - 3(6 + 2)\)

Step 2: Simplify Both Sides of the Equation

Simplify both sides of the equation to solve it. \(2(3) - 17 = 13 - 3(8)\) which gives \(-11 = -11\)

Step3: Check if the equation balances

Since \(-11 = -11\), the equation is balanced, thus it indicates that 6 is indeed a solution to this equation.

Key Concepts

Substitution Method in Algebra

The substitution method is a fundamental technique used to solve algebraic equations, particularly when dealing with one or more unknown variables. This method involves replacing a variable with its respective numerical value or another expression to simplify the equation and find the solutions.

For example, take the simple equation given in the original exercise: to determine whether 6 is a solution for the equation \(2(x-3)-17=13-3(x+2)\), we begin by substituting the variable \(x\) with the number 6. This initial step is crucial as it transforms the abstract equation into a concrete numeric expression that we can analyze further.

Equation Simplification

After substituting the variable with a value, the next step is to simplify the equation. Simplification involves performing arithmetic operations like addition, subtraction, multiplication, and division to condense the equation to its most basic form. This often includes expanding brackets and combining like terms.

In the context of our given problem, once we substitute 6 for \(x\), we simplify each side of the equation to find that \(2\times(3) - 17 = -11\) and \(13 - 3\times(8) = -11\). The simplification step serves to reveal whether the left and right sides of the equation are indeed equal, which is the essence of finding a solution to an equation.

Verifying Solutions of Equations

Verifying the solution of an equation is a critical step in the problem-solving process. It involves making sure that the left-hand side (LHS) and the right-hand side (RHS) of the equation are equal when the suspected solution is substituted into the original equation.

In our original exercise, after simplifying, we see that the LHS and RHS both equal -11. This equality confirms that our initial substitution was valid, and hence, 6 is indeed a solution to the equation \(2(x-3)-17=13-3(x+2)\). Verification serves as a proof that solidifies our confidence in the accuracy of the solution.