Problem 41 - Algebra 1

Question

Write a question that can be used to solve the equation. Then use mental math to solve the equation. \(2 b-1=10\)

Step-by-Step Solution

Answer

So, the solution to the equation is \(b = 5.5\).

Step 1: Translate the Equation

Given the equation \(2 b-1=10\), the corresponding question could be: 'What number \(b\), when multiplied by 2 and then subtracted by 1, gives 10?'

Step 2: Isolate the Variable

Start by manipulating the equation to isolate \(b\). To do that, add 1 to both sides of the equation resulting in: \(2b = 10 + 1\), which simplifies to: \(2b = 11\).

Step 3: Solve for Variable

Divide both sides of the equation by 2 to solve for \(b\). The equation now becomes: \(b = 11 / 2\).

Key Concepts

Mental Math

Solving equations can sometimes appear challenging, but mental math can simplify the process. For the equation \(2b - 1 = 10\), we start by understanding the operations involved. The equation tells us to multiply a number \(b\) by 2, then subtract 1 to get 10. To solve this mentally, think backward.

• Add 1 to 10, which results in 11. This undoes the subtraction of 1 in the original equation.
• Now, consider what number, when multiplied by 2, gives 11. Divide 11 by 2 to find \(b\).

By quickly reversing the operations in your mind, you arrive at \(b = 5.5\). Mental math tricks like these are particularly useful because they enhance your problem-solving skills without pen and paper.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to make them easier to solve. In our equation, \(2b - 1 = 10\), the goal is to make \(b\) the subject.
This means modifying the equation so that \(b\) is isolated on one side. This often involves reversing operations: if something is subtracted, you add; if multiplied, you divide.

• Start by adding 1 to both sides: \(2b = 11\). This cancels out the \(-1\) next to \(b\).
• Now, we need to deal with the 2 next to \(b\). Since \(b\) is multiplied by 2, divide the whole equation by 2 to isolate \(b\).

The equation simplifies directly to \(b = 5.5\). Each step of manipulation simplifies the equation further, eventually leading to the solution for \(b\). Algebraic manipulation is like a toolkit, enabling you to handle equations of varying complexity.

Isolation of Variables

The isolation of variables is a core skill in solving equations. This process involves getting the variable of interest, \(b\) in our case, by itself on one side of the equation.

• The first step is to simplify the equation: add 1 to both sides, resulting in \(2b = 11\).
• Next, address the coefficient of \(b\), which is 2 here. Divide both sides by 2 to solve for \(b\): \(b = 11 / 2\).

The end result is \(b = 5.5\), a simple number uncovered by careful isolation. Mastering this skill is essential in algebra as it forms the foundation for dealing with much larger systems of equations. By practicing isolation, students gain confidence in systematically solving equations, regardless of their complexity.