Problem 36 - Algebra 1

Question

Solve the equation if possible. $$ 9(b-4)-7 b=5(3 b-2) $$

Step-by-Step Solution

Answer

The solution to the equation is \(b = -2\).

Step 1: Distribute the Multiplication

Begin by distributing the multiplication through the parentheses in the equation: \(9b - 36 - 7b = 15b -10\).

Step 2: Combine Like Terms on Each Side

Combine like terms on both sides of the equation, which will yield the simplified equation: \(2b - 36 = 15b - 10\)

Step 3: Isolate the Variable

Attempt to isolate 'b' to one side of the equation. This can be done by subtracting \(2b\) from both sides and adding \(10\) to both sides, which gives you: \(-36 + 10 = 15b - 2b\).

Step 4: Simplify

Now simplify again by combining like terms: \(-26 = 13b\).

Step 5: Solve for the Variable

Lastly, solve for 'b' by dividing both sides of the equation by 13: \(-26/13 = b\) yields \(b = -2\).

Key Concepts

Understanding Distributive Property

When faced with an equation that includes terms inside parentheses, like in our exercise, the distributive property is your friend. This property allows you to multiply a single term by each term inside the parentheses, making it simpler to work with. In our equation, we apply the distributive property to both sides: on the left, you have \(9(b-4)\) which gets distributed to become \(9b - 36\). On the right, \(5(3b-2)\) becomes \(15b - 10\). This simplification step is crucial because it helps eliminate the parentheses, making the equation easier to manage.

Mastering the Art of Combining Like Terms

Once you've distributed the multiplication across the parentheses, your next goal is to simplify the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In the equation from our exercise, after distribution, we notice both sides have terms with the variable \(b\): \(9b\) and \(-7b\) on the left, and \(15b\) on the right. Combine these to simplify the equation to \(2b - 36 = 15b - 10\).

• Look for terms with the same variables.
• Add or subtract their coefficients.

By combining like terms efficiently, you reduce the complexity of the equation, paving the way for easier variable isolation.

Steps to Variable Isolation

The final hurdle in solving an equation like this one is to isolate the variable, meaning you want the variable on one side of the equation and the constants on the other. Start by moving all the terms with the variable to one side. Subtract \(2b\) from both sides, you get \[-36 = 13b - 10\]. Then, add \(+10\) to both sides so that all constants are on one side, leading to \(-26 = 13b\). Here's a quick guide to isolation:

• Use addition or subtraction to move terms without the variable across the "=" sign.
• Use division or multiplication to solve for the variable.

Finally, divide every term by 13 to solve for \(b\), yielding \(b = -2\). By isolating the variable, you clearly determine its value, completing the equation-solving process.