Problem 131 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether 2 is a solution of \(13 x+3=3(5 x-1)\)

Step-by-Step Solution

Answer

No, 2 is not a solution to the given equation.

Step 1: Substitute the value of x with 2

The equation given is \(13 x+3=3(5 x-1)\). Substitute \(x = 2\) in the equation to get \(13*2 + 3 = 3(5*2 - 1)\) which simplifies to 29 = 27.

Step 2: Check whether both sides of the equation are equal

Check whether the two sides of the equation are equal. If they are, then 2 is a solution. In this case, 29 ≠ 27. Therefore, the value 2 is not a solution to the equation.

Key Concepts

Algebraic Solutions

As part of understanding algebra, figuring out {\bf algebraic solutions} is a fundamental skill. An algebraic solution is essentially a value (or set of values) that makes an equation true. When you're given an equation like the one in our exercise \(13x + 3 = 3(5x - 1)\), to confirm if a number is a solution, you substitute it into the equation. This means you replace the variable, in this case \(x\), with the number you’re testing – here it's 2.

After the substitution, if both sides of the equation balance out, meaning they are equal, then you’ve found a correct solution! If not, the number is not a correct solution for the equation. It's like testing if a key fits a lock; only the right key (solution) will unlock (balance) it. The equation from our exercise, when we substituted 2 in for \(x\), gave us a result of 29 = 27, which is not true, indicating that 2 is not the right 'key' for this 'lock'.

Substitution Method

The {\bf substitution method} is a technique we use to solve algebraic equations, particularly systems of equations. The method involves replacing a variable with an equivalent expression from another equation to solve for the variables. However, even for a single equation, like \(13x + 3 = 3(5x - 1)\), this method can be employed to verify if a certain number is a solution.

The process goes like this: you first identify the variable to be substituted, then replace it with a given value, and finally, simplify to see whether the substitution leads to a true statement. It's akin to swapping out puzzle pieces to see if they perfectly fit into place. In our case, the substitution \(x = 2\) did not fit because it resulted in a false equation, hence 2 is not a valid piece of this particular puzzle.

Equivalence of Equations

The concept of {\bf equivalence of equations} revolves around two or more equations representing the same relationship. In simpler terms, equivalent equations have the same set of solutions. You can often create them by manipulating an original equation using algebraic operations like adding or multiplying both sides by a number. It's key to remember that whatever you do to one side, you must do to the other to maintain the balance.

When we check for the solution of an equation, we're trying to maintain this tricky balance. If after substituting a value into an equation, the two sides remain equal, the equations are equivalent with respect to that particular value, signifying it as a solution. In the exercise with the equation \(13x + 3 = 3(5x - 1)\), we sought to determine if the equation would remain equivalent when \(x = 2\). It turned out that substituting 2 for \(x\) broke the equivalence, proving that 2 isn't a solution for the given equation.