Problem 26 - Prealgebra

Question

Solve equation. Check your solution. \(2 b+6.2=13.2-8 b\)

Step-by-Step Solution

Answer

The solution is \(b = 0.7\).

Step 1: Move all terms involving 'b' to one side

Start by adding \(8b\) to both sides of the equation to get terms involving 'b' on one side: \(2b + 8b + 6.2 = 13.2\).

Step 2: Simplify the equation

Combine like terms: \(10b + 6.2 = 13.2\).

Step 3: Isolate the term with 'b'

Subtract 6.2 from both sides to isolate the term with 'b': \(10b = 13.2 - 6.2\).

Step 4: Simplify the right side

Calculate the subtraction on the right side: \(10b = 7\).

Step 5: Solve for 'b'

Divide both sides by 10 to solve for 'b': \(b = \frac{7}{10}\).

Step 6: Check your solution

Substitute \(b = 0.7\) back into the original equation: \(2(0.7) + 6.2 = 13.2 - 8(0.7)\). Calculate both sides: \(1.4 + 6.2 = 13.2 - 5.6\). Both sides equal 7.6, so the solution is correct.

Key Concepts

Understanding Algebra Basics

Algebra can seem puzzling at first, but it's essentially about finding the value of unknown variables like \(b\) in equations. These equations are mathematical statements that show the equality of two expressions using an equals sign. To solve for a variable, follow these key steps:

• Move all terms with the variable you want to solve for, in this case 'b', to one side of the equation. This allows you to focus on isolating the variable.
• Convert any operations made on the variable. For instance, if something is added to the variable on its side, you'll need to subtract that same value from the other side.
• Finally, simplify each side, ensuring your variable is on its own and other constants are isolated as needed.

These steps help in maintaining the balance of the equation while isolating the variable you need to solve for.

Checking Solutions

Once you find a potential solution, it's crucial to check its accuracy. This involves substituting your calculated value back into the original equation to see if it satisfies both sides. For instance, with \(b = 0.7\):

• Substitute \(b = 0.7\) back into the equation: \(2(0.7) + 6.2 = 13.2 - 8(0.7)\).
• Calculate both sides separately. Start with the left side: \(2 \times 0.7 + 6.2 = 1.4 + 6.2 = 7.6\).
• Then calculate the right side: \(13.2 - 8 \times 0.7 = 13.2 - 5.6 = 7.6\).
• Since both sides equal 7.6, the solution \(b = 0.7\) is verified as correct.

Checking is a fundamental part of solving equations, as it confirms whether your solution is valid in the context of the original mathematical statement.

Simplifying Equations

Simplifying an equation involves combining and reducing terms to make the equation easier to solve. Your goal is to have as few terms as possible, ideally only those involving the unknown variable on one side. Here's how:

• Combine like terms, which are terms with the same variable component. For instance, \(2b + 8b\) simplifies to \(10b\).
• Perform arithmetic operations to move constants to the opposite side of the variable. In this example, subtracting 6.2 from both sides moves the constant, simplifying to \(10b = 7\).
• If further simplification is required, divide or multiply to get the variable alone, leading you to the final solution, such as \(b = \frac{7}{10}\) or \(0.7\).

Simplifying equations efficiently makes it easier to find and verify solutions by reducing the clutter and focusing attention on the variable in question.