Problem 34 - Elementary Algebra

Question

Is the given value a solution to the linear equation? $$ 4 x-3=-3 x ; x=-2 $$

Step-by-Step Solution

Answer

The given value \(x = -2\) is not a solution.

Step 1: Substitute the Value

Plug the given value of \(x = -2\) into the equation \(4x - 3 = -3x\). This will allow us to check if the left-hand side equals the right-hand side when \(x = -2\).

Step 2: Calculate the Left Side

Substitute \(x = -2\) into the left side of the equation: \[4(-2) - 3 = -8 - 3 = -11.\]

Step 3: Calculate the Right Side

Now, substitute \(x = -2\) into the right side of the equation: \[-3(-2) = 6.\]

Step 4: Compare Both Sides

Compare the results of both sides. On the left, we have -11, and on the right, we have 6. These are not equal.

Key Concepts

Solution Verification

Verifying whether a given value is a solution to a linear equation is a crucial concept in algebra. When we are tasked with determining if a specific value, like \(x = -2\), satisfies a linear equation, we engage in solution verification. The goal is to see whether both sides of the equation become equal when substituting the given value into the equation.
Here's how it works:

• Substitute the given value into the equation.
• Calculate both sides of the equation separately.
• Compare the two results to see if they match.

In our original exercise, after substitution, the left side resulted in \(-11\) and the right side resulted in \(6\), showing us that \(x = -2\) was not a valid solution. This process helps confirm or deny if a particular value solves the equation correctly.

Substitution Method

The substitution method is a straightforward technique used in solving algebraic equations, including linear ones. It involves replacing variables in an equation with specific values to simplify the equation or verify solutions.
In the context of solution verification, you do this to check if a given value is a solution to the equation:

• Identify the variable in the equation. For our example, it's \(x\).
• Insert the given value into every instance of that variable in the equation.
• Calculate each side of the equation to see if they are equal.

This method is not only useful for confirming solutions but also for simplifying equations when solving systems of equations. By efficiently substituting values, this method provides clarity and direction when working through algebraic equations.

Algebraic Equations

Algebraic equations are mathematical statements that use variables and constants to express relationships. They form a backbone of algebra and come in various types such as linear, quadratic, and polynomial equations. Linear equations, like in our exercise, involve variables to the first degree, and they graph as straight lines.
Understanding algebraic equations is essential:

• They use coefficients, variables, and constant terms.
• They can be manipulated using mathematical operations like addition, subtraction, multiplication, and division.
• They serve as tools for modeling real-world situations and solving problems.

The essence of solving algebraic equations, like verifying a solution as in our exercise, lies in equalizing both sides through defined operations. Mastery of these equations opens the door to deeper mathematical concepts and real-life applications.