Problem 8 - Elementary Algebra Within Reach

Question

In Exercises \(1-10\), determine whether each value of \(x\) is a solution of the equation. \(5 x-1=3(x+5)\) (a) \(x=8\) (b) \(x=-2\)

Step-by-Step Solution

Answer

Based on the solution steps, \(x = 8\) is a solution to the equation, but \(x = -2\) is not a solution.

Step 1: Substitute \(x = 8\) into the equation

Substitute \(x = 8\) into the equation, we get:\[5(8) - 1 = 3(8 + 5)\]Which simplifies to:\[39 = 39\]So, \(x = 8\) is a solution to the equation.

Step 2: Substitute \(x = -2\) into the equation

Now, substitute \(x = -2\) into the equation, and we have:\[5(-2) - 1 = 3(-2 + 5)\]Simplifying this gives:\[-11 = 9\]They are not the same, so \(x = -2\) is not a solution to the equation.

Key Concepts

Equation Solving

Equation solving is a foundational concept in mathematics, particularly algebra, where the objective is to find the value(s) of variables that make a given equation true. In the context of the exercise presented, the goal was to verify if certain values of the variable x satisfy the equation 5x - 1 = 3(x + 5). To solve such linear equations, we typically look for the x that will balance the equation, meaning both sides of the equal sign will have the same value.

Determining whether a value is a solution involves substituting that value into the equation and simplifying both sides to see if they are equal. If upon simplification, the equation holds true (the left side equals the right side), then that value is a solution; if not, it isn't. Such skills are essential as they form the basis for more complex mathematical problem-solving, including in fields such as engineering, science, and economics.

Substitution Method

The substitution method is a technique used to solve equations, where one replaces a variable with its value to determine whether the equality stands. This approach is effective for assessing if a specific number is a solution to an equation. In the given exercise, students were asked to apply this method by substituting given values of x (8 and -2) into the equation.

Substitution makes it straightforward to see the logical sequence involved in verifying a solution. After substitution, simplifying the expressions on both sides provides a clear yes or no answer to the question of whether the given number solves the equation. The method also reinforces the understanding of mathematical operations and their properties.

Algebraic Solutions

Algebraic solutions refer to the values obtained from solving algebraic equations that involve variables. These solutions are critical as they represent the possible answers to problems posed in an algebraic format. In the provided exercise, there were specific potential solutions to the linear equation – 8 and -2 – and they were tested using the substitution method. An algebraic solution is deemed correct when, upon substitution, the original equation is satisfied.

Fostering the ability to find algebraic solutions is not just about knowing the techniques, but also about understanding the underlying concepts, such as variables, operations, and the importance of maintaining the balance of an equation. It is this foundational comprehension that enables students to tackle a wide variety of algebraic problems and apply their skills in practical, real-world scenarios.