Problem 122 - Introductory and Intermediate Algebra for College Students 4th

Question

In each exercise, determine whether the given number is a solution of the equation. $$4 x+2=3(x-6)+8 ;-11$$

Step-by-Step Solution

Answer

-11 is not a solution to the equation \(4x + 2 = 3(x - 6) + 8\).

Step 1: Substitute the given number into the equation

Plug in -11 in place of \(x\) in the equation: \(4*(-11) + 2 = 3*(-11 - 6) + 8\).

Step 2: Simplify both sides

Simplify the left and the right side of the equation separately. For the left side: \(4*(-11) + 2 = -44 + 2 = -42\). For the right side: \(3*(-11 - 6) + 8 = 3*(-17) + 8 = -51 + 8 = -43\).

Step 3: Compare the results

Now compare both simplified sides of the equation. They are -42 and -43, which do not equal each other. This means, -11 is not a solution to this equation.

Key Concepts

Substitution Method

The substitution method is a cornerstone of algebra and is used to determine whether a number is a solution to an equation. It involves replacing a variable with a specific value and performing the operations to assess the validity of the substitution.

To use this method, you should first identify the variable in an equation. In our exercise, the variable is represented by 'x'. The next step is to substitute the proposed solution—in this case, -11—into the equation wherever 'x' appears. This process turns the abstract equation into a concrete number sentence that can be evaluated.

Remember, the substitution is only the first part; the crucial step is to simplify the resulting expressions to see if both sides of the equation balance. If they do, you've found a valid solution. If not, as in our exercise, the substituted number is not a solution to the equation.

Equation Simplification

Equation simplification is the process of breaking down complex equations into simpler forms to make them easier to solve. This process often includes combining like terms, distributing multiplication over addition or subtraction, and performing arithmetic operations.

In our exercise example, simplification occurs in two stages, one on each side of the equation:

• On the left, multiplying 4 by -11 and adding 2 to simplify to -42.
• On the right, expanding the parentheses by multiplying 3 with (-11 - 6), and then adding 8, which simplifies to -43.

Simplification is a powerful tool that can reveal the solution to many algebraic problems or, as with our example, show that a proposed solution is incorrect. It's essential to perform each operation step by step and avoid making errors that could change the equation's meaning.

Algebraic Problem Solving

Algebraic problem solving is a systematic approach to finding solutions to mathematical problems using algebraic techniques. It involves understanding what is asked in the problem, identifying unknowns, and applying methods such as the substitution method and simplification, just as we saw in our textbook exercise.

The problem-solving process usually consists of several stages:

• Understanding the problem: Take the time to analyze what is being asked and what the given information is.
• Devising a plan: Decide which algebraic methods will best address the question. Is substitution appropriate, or would another method work better?
• Carrying out the plan: Implement the chosen method(s) step by step, ensuring each operation is performed accurately.
• Reviewing/Reflecting: Verify whether the solution makes sense and whether all parts of the problem have been addressed.

In the provided exercise, we used algebraic problem solving to determine that -11 is not a solution to the given equation. Each step in the process is crucial to reaching the correct conclusion.