Question

First solve the problem, and then enter your answer on the grid provided on the answer sheet. The instructions for entering your answers follow. If \(2 x+3=4\), what is the value of \(6 x+9 ?\)

Short answer

The value of \(6x + 9\) is 12.

Step-by-step solution

Solve for x

To find the value of \(x\), solve the given equation \(2x + 3 = 4\). Subtract 3 from both sides to get \(2x = 1\). Then, divide both sides by 2 to solve for \(x\): \(x = \frac{1}{2}\).

Substitute x into the Expression

Now that we know \(x = \frac{1}{2}\), substitute it into the expression \(6x + 9\). This becomes \(6 \times \frac{1}{2} + 9\).

Calculate the Expression

First, calculate \(6 \times \frac{1}{2}\), which equals 3. Then substitute this back into the expression: \(3 + 9\).

Final Result

Add the numbers together: \(3 + 9 = 12\). Thus, the value of \(6x + 9\) is 12.

Key concepts

Understanding Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and expressions. It allows us to express mathematical ideas and relationships in a general way, making it essential for solving problems efficiently. Instead of using specific numbers, algebra uses variables, like \(x\) or \(y\), which stand in for unknown values. This flexibility is what makes algebra so powerful in mathematics.

In our problem, we use algebra to solve for \(x\) in the equation \(2x + 3 = 4\). The goal is to isolate \(x\), meaning that we want to get \(x\) by itself on one side of the equation. This process involves techniques such as addition, subtraction, multiplication, and division, which are cornerstones of algebra.

Solving Linear Equations

Linear equations are equations of the first degree, which means they have variables that are not squared, cubed, or raised to any higher power. They're called linear because if you graphed them, they would form a straight line. A typical linear equation looks like \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.

Here's how to solve a linear equation step by step, using our example \(2x + 3 = 4\):

• Start by isolating the variable term. This involves removing any other numbers that are added or subtracted from it. In the equation \(2x + 3 = 4\), subtract 3 from both sides to get \(2x = 1\).
• Next, solve for \(x\) by doing the opposite of multiplying by 2, which is dividing by 2. This gives us \(x = \frac{1}{2}\).

Mastering these steps enables you to tackle more complex equations confidently. Once \(x\) is isolated, we've solved the equation!

Working with Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (like +, -, \(\times\), and \(\div\)) that represent a value or concept. In the context of our exercise, the expression we focused on was \(6x + 9\).

To evaluate a mathematical expression like \(6x + 9\) when \(x\) is known, follow these steps:

• First, substitute the known value of the variable into the expression. Here, replace \(x\) with \(\frac{1}{2}\) to get \(6 \times \frac{1}{2} + 9\).
• Then, carry out the operations in the expression. First, calculate the multiplication: \(6 \times \frac{1}{2} = 3\).
• Finally, complete the expression by adding the remaining numbers: \(3 + 9 = 12\).

This approach helps clarify and simplify complex expressions, making them manageable and providing a clear path to the solution. By understanding how to work with expressions, you're better equipped to interpret and solve a wide range of mathematical problems.