Solve for b 12(b+2)=8(b+5)

The question is asking for the value of the variable 'b' that satisfies the equation. The equation provided is a linear equation in one variable, where the variable 'b' is present on both sides of the equality sign. The equation expresses a relationship where 12 times the sum of 'b' and 2 is equal to 8 times the sum of 'b' and 5. The problem requires manipulation of algebraic expressions, such as distributing the multiplication across the parentheses, combining like terms, and isolating the variable on one side to find the value of 'b' that makes the equation true.

$12 (b + 2) = 8 (b + 5)$

Answer

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Solution:

Step:1

Expand the expression $12 (b + 2)$ .

Step:1.1

Express the equation as $0 + 0 + 12 (b + 2) = 8 (b + 5)$ .

Step:1.2

Include zero additions for clarity: $12 (b + 2) = 8 (b + 5)$ .

Step:1.3

Use the distributive law: $12 b + 12 \cdot 2 = 8 (b + 5)$ .

Step:1.4

Perform the multiplication: $12 b + 24 = 8 (b + 5)$ .

Step:2

Expand the expression $8 (b + 5)$ .

Step:2.1

Use the distributive law: $12 b + 24 = 8 b + 8 \cdot 5$ .

Step:2.2

Perform the multiplication: $12 b + 24 = 8 b + 40$ .

Step:3

Isolate terms with $b$ on one side.

Step:3.1

Subtract $8 b$ from both sides: $12 b + 24 - 8 b = 40$ .

Step:3.2

Combine like terms: $4 b + 24 = 40$ .

Step:4

Isolate terms without $b$ on the other side.

Step:4.1

Subtract $24$ from both sides: $4 b = 40 - 24$ .

Step:4.2

Calculate the difference: $4 b = 16$ .

Step:5

Divide to solve for $b$ .

Step:5.1

Divide the equation by $4$ : $\frac{4 b}{4} = \frac{16}{4}$ .

Step:5.2

Simplify the left side.

Step:5.2.1

Eliminate the common factor of $4$ .

Step:5.2.1.1

Reduce the fraction: $\frac{4 b}{4} = \frac{16}{4}$ .

Step:5.2.1.2

Simplify to $b$ : $b = \frac{16}{4}$ .

Step:5.3

Simplify the right side.

Step:5.3.1

Divide $16$ by $4$ : $b = 4$ .

Knowledge Notes:

This problem involves solving a linear equation, which is an equation of the first degree, meaning it contains only the first power of the variable. The steps taken to solve the equation are based on the following algebraic principles:

Distributive Property: This property states that $a (b + c) = a b + a c$ . It is used to expand expressions that involve multiplication over addition or subtraction.
Combining Like Terms: Terms that have the same variable raised to the same power can be combined by adding or subtracting their coefficients.
Isolation of the Variable: To solve for a variable, we need to isolate it on one side of the equation. This involves moving terms with the variable to one side and constants to the other, using addition or subtraction.
Simplification: After isolating the variable, we often need to simplify the equation by dividing both sides by the coefficient of the variable to find its value.
Checking the Solution: Although not explicitly shown in the steps, it's always a good practice to check the solution by substituting the value of the variable back into the original equation to ensure that both sides of the equation are equal.