Problem 100 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. When I substituted 5 for \(x\) in the equation $$4 x+6=6(x+1)-2 x$$ I obtained a true statement, so the equation's solution is 5 .

Step-by-Step Solution

Answer

The statement 'When I substituted 5 for \(x\) in the equation \(4x+6 = 6(x+1)-2x\), I obtained a true statement, so the equation's solution is 5' does make sense because substituting \(x = 5\) in the equation leads to an equality.

Step 1: Substitution

Begin by substituting \(x = 5\) into the given equation \(4x + 6 = 6(x + 1) - 2x\). It becomes \(4(5)+6 = 6((5)+1)-2(5)\) that simplifies to \(20+6 = 6(6)-10\).

Step 2: Simplify

Next, perform the operations to further simplify the equation. The left side becomes \(26\) and the right side simplifies to \(36-10 = 26\).

Step 3: Comparison

Finally, compare both sides of the equation to see if they are equal. It is observed that both sides are indeed equal i.e. \(26 = 26\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra that involves replacing a variable with its corresponding numerical value. This process helps to simplify an equation and verify if a given value is a solution. In practice:

• You start with an equation and a value that you want to check.
• Replace the variable in the equation with the given number.
• Simplify the equation to see whether you get a true statement, such as a matching equality on both sides.

In our exercise, when 5 is substituted for the variable x in the equation, it transforms from an algebraic expression into a simple arithmetic one. This makes it clear and tangible whether the original equation holds true for x equals 5.

Solving Algebraic Equations

To solve an algebraic equation, you are essentially finding the value(s) for the variable(s) that make the equation true. You can approach this in several ways, depending on the complexity of the equation. For linear equations, common methods include graphing, using the substitution or elimination methods, or rewriting in function form.

In the case of our exercise using the substitution method, we begin by inserting the value of x into the equation. We then simplify to find out if the value of x solves the equation. If after the simplification, both sides of the equation have an equal numerical value, the chosen value for x is a true solution to the equation.

Equation Simplification

Equation simplification involves reducing an algebraic equation to its simplest form, making it easier to interpret or solve. Think of simplification as cleaning up the mathematics by combining like terms, factoring, or executing algebraic operations that result in a less complicated expression.

• Simplify expressions on each side of the equation by performing arithmetic operations.
• Combine like terms and reduce expressions to the lowest terms possible.
• Keep the equation balanced, making identical modifications to both sides if necessary.

During the simplification step in our exercise, arithmetic operations reduce the original equation to a much clearer statement, revealing that both sides equal 26. This simplicity aids in effective comparison and reduces the likelihood of errors in evaluating whether the given value for x is correct.

Validating Solutions

Validating solutions means verifying that your answers actually satisfy the original equations. To validate a solution:

• Take the proposed solution and substitute it back into the original equation.
• Simplify the equation as was done in the simplification step.
• If both sides of the simplified equation are equal, your solution is validated.
• If not, either the proposed solution is incorrect, or there may have been an error during the simplification or substitution process.

In the exercise, after simplifying the equation with the substituted value, we compare both sides for equality. Since they are equal, we have validated that the solution x equals 5 fits appropriately in our equation, reinforcing the confidence in our answer.