Problem 15 - Algebra 2

Question

STANDARDIZED TEST PRACTICE If \(4 x+7=18,\) what is the value of \(12 x+21 ?\) \(\begin{array}{llll}{\text { A } 2.75} & {\text { B } 32} & {\text { C } 33} & {\text { D } 54}\end{array}\)

Step-by-Step Solution

Answer

The value of \(12x + 21\) is 54, so the answer is D: 54.

Step 1: Solve for x

To find the value of \(x\), we start from the equation given: \(4x + 7 = 18\). We subtract 7 from both sides to isolate the \(4x\) term: \(4x = 18 - 7\), which simplifies to \(4x = 11\). Then, we divide both sides by 4 to solve for \(x\): \(x = \frac{11}{4}\).

Step 2: Substitute x into the new expression

Now, substitute \(x = \frac{11}{4}\) into the expression \(12x + 21\). This yields \(12\left(\frac{11}{4}\right) + 21\).

Step 3: Simplify the expression

First, compute \(12 \times \frac{11}{4}\): \(12 \times \frac{11}{4} = \frac{132}{4} = 33\). Then add 21: \(33 + 21 = 54\).

Step 4: Choose the correct answer

From the simplification in Step 3, \(12x + 21 = 54\). Therefore, the correct answer is D: 54.

Key Concepts

Substitution Method

The substitution method is a useful technique for solving equations. It's a way to easily tackle algebraic problems by "substituting" known values back into an equation. In simpler terms, once you find the value of a variable, you can plug that value into another expression to find the solution. In our linear equation exercise, we first solved for the unknown variable \(x\) as \(\frac{11}{4}\). Then, we "substituted" \(x\) into the new expression \(12x + 21\).

This substitution allowed us to evaluate the expression using concrete numbers rather than abstract symbols, simplifying our work. Here's how substitution helped:

• It eliminated the unknown variable by replacing it with a known value.
• It facilitated the calculation of additional expressions involving the variable.

Hence, even complex expressions can become manageable with this method by providing a straightforward pathway to get the solution.

Algebraic Expressions

Algebraic expressions like \(12x + 21\) are fundamental in mathematics. An algebraic expression consists of numbers, variables, and operations such as addition, division, etc. It's a way to represent quantities and calculate new values. For the exercise, the expression \(12x + 21\) was derived to find a specific value by substituting \(x = \frac{11}{4}\).

Key parts of an algebraic expression include:

• Coefficients: Numbers in front of variables. In our exercise, \(12\) is the coefficient of \(x\).
• Constants: Stand-alone numbers, like \(21\) in the expression.
• Variables: Symbols representing unknown numbers, \(x\) in this case.

Recognizing these components helps decipher the expression more effectively. Algebraic expressions allow us to encapsulate real-world situations mathematically, providing flexibility and precision in problem-solving.

Problem Solving Steps

Breaking down a math problem into solvable steps can make it much easier to manage. In our exercise, we had several clearly defined problem-solving steps. Let's review them:

• First, we solve for the variable \(x\) by isolating it in the given equation \(4x + 7 = 18\).
• Next, we substituted the result into another expression, \(12x + 21\), to solve for the final value.
• Finally, we simplified the expression step by step to deduce \(54\), which matched the answer option D.

In each step, we targeted a specific task, whether isolating the variable or performing arithmetic operations. Structured approaches guide us through even the trickiest problems, ensuring that each part of the solution builds upon the previous, ultimately leading to the correct answer. Being methodical not only gets us to the solution but boosts our understanding and confidence!