Question

Write a question that can be used to solve the equation. Then use mental math to solve the equation. \(3 x+15=24.6\)

Short answer

The value of \(x\) in the equation \(3x + 15 = 24.6\) is 3.2.

Step-by-step solution

Formulating the Question

The equation \(3x + 15 = 24.6\) can be formulated into the question: If \(3x + 15\) equals 24.6, what is the value of \(x\)?

Isolating the 'x' term

To solve for \(x\), the aim is to isolate \(x\) on one side of the equation. Start by moving 15 to the other side of the equation. This could be done by subtracting 15 from both sides. So, the equation becomes \(3x = 24.6 - 15\).

Solving for 'x'

Perform the subtraction operation on the right side of the equation, then divide each side by 3 to find the value of \(x\). Thus, \(x = (24.6 - 15) / 3\).

Calculating the Result

After performing these operations, \(x\) equates to 3.2.

Key concepts

Mental Math

When faced with a linear equation such as 3x + 15 = 24.6, applying mental math can expedite the process of finding the solution. Mental math involves doing calculations in your head, using number sense and familiarity with numbers to simplify and solve problems. In our example, you could recognize that subtracting 15 from 24.6 leaves you with 9.6. Since 3 multiples to give 9.6 aren't immediately apparent, you could use estimation and adjustment techniques. Knowing that 3 times 3 is 9 and adjusting for the decimal can guide you to deduce that 3 times 3.2 is close to 9.6. This way, mental math not only saves time but also enhances your numerical understanding and agility.

Isolating Variables

Isolating variables is a foundational skill in solving algebraic equations. It involves manipulating the equation to get the unknown variable, often denoted as x, by itself on one side of the equality sign.

The goal is to reverse the operations that have been applied to the variable. In our exercise 3x + 15 = 24.6, the variable x is first subjected to multiplication by 3 and then an addition of 15. To isolate x, you perform the inverse of these operations in reverse order, which means you would subtract 15 from both sides, and then divide by 3. This systematic approach helps to clear out the numbers and operations surrounding the x, making the solution for the variable apparent.

Algebraic Expressions

Algebraic expressions represent a combination of numbers, variables, and arithmetic operations without an equality sign. In contrast, algebraic equations include an equality sign and typically represent a problem to be solved. Understanding algebraic expressions is crucial because they are the building blocks of equations.

For example, in the expression 3x + 15 from the equation 3x + 15 = 24.6, you see multiplication and addition at play. Recognizing these operations allows you to effectively reverse them when solving the equation. By mastering the manipulation of algebraic expressions, you gain the confidence to tackle complex problems and develop your analytical skills.