Question

MENTAL MATH Write a question that could be used to solve the equation. Then use mental math to solve the equation. $$4 r-1=11$$

Short answer

The solution to the equation is \(r = 3\).

Step-by-step solution

Understand the Equations and Requirements

First, understand the equation \(4r-1=11\). This equation implies 4 times some number \(r\) minus 1 is equal to 11. The task here is to find the number \(r\).

Isolate the variable

Next step is to isolate the variable \(r\) by getting rid of everything else around it. Start by getting rid of -1 in the equation by adding 1 to both sides. This gives \(4r = 11 + 1\).

Simplify the equation

Simplify the right-hand side of the equation by adding up the numbers. This results in the new equation \(4r = 12\).

Solve for the variable

Lastly, solve for \(r\) by dividing both sides of the equation by 4. This gives \(r = 12 / 4\).

Final Simplification

When 12 is divided by 4, the result is 3. So, \(r = 3\), which is the solution to the equation.

Key concepts

Mental Math

Mental math is the skill of performing calculations in one's head without the aid of any writing materials, calculators, or other tools. It's not just a test of computation skills, but also a measure of your understanding of number properties and relationships. This skill is particularly valuable when solving linear equations, as it allows you to quickly assess and manipulate numbers to drive toward a solution.

For instance, with the equation \(4r - 1 = 11\), you could easily do mental math to add 1 to both sides, knowing that \(11 + 1\) equals 12. It's an intuitive process that can often shorten the amount of time needed to solve problems and is especially useful for equations designed to be solved this way. In educational settings, fostering mental math skills can help students enhance their mathematical fluency and intuition.

Isolating the Variable

One of the fundamental steps in solving linear equations is the concept of isolating the variable. This means rearranging the equation so that the variable we want to solve for is on one side of the equation, and everything else is on the other side. When dealing with simple linear equations, the goal is to have the variable by itself, allowing us to see what it equals when compared to the number on the other side of the equals sign.

Considering our example \(4r - 1 = 11\), your goal would be to isolate \(r\). This is achieved by performing the same operation on both sides of the equation to maintain its balance. By adding 1 to each side, you are effectively moving the -1 over to the other side, now having \(4r = 12\). The equation is now set up for the variable \(r\) to be easily found, demonstrating the power and necessity of isolating the variable in solving linear equations.

Simplification of Equations

Simplification of equations is a process of turning a complex mathematical expression into the simplest form possible by combining like terms and eliminating unnecessary parts of the equation. Simplification can involve arithmetic operations such as addition, subtraction, multiplication, and division, and is key to making equations more solvable and understandable.

In the example \(4r = 12\), we simplify by dividing each side by four, which is the coefficient of \(r\). This step reduces the equation to its most straightforward form: \(r = 3\). Through simplification, the core relationship between the variables becomes clear, and this final result manifests the essence of the equation without extraneous elements. Mastering simplification processes can greatly improve a student's efficiency and confidence when tackling various algebraic problems.