Problem 46 - Algebra 1

Question

Solve the equation. $$4 b=26-9 b$$

Step-by-Step Solution

Answer

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

Step 1: Rearranging the Equation

The first step is to rearrange the equation to bring all terms involving \(b\) on one side. This can be done by adding \(9b\) to both sides of equation. This gives us \(4b + 9b = 26\).

Step 2: Simplify the Left Hand Side

Adding the terms on the left hand side will simplify the equation. This results in \(13b = 26\).

Step 3: Solve for Variable b

Now, to solve for \(b\), divide both sides of the equation by \(13\). This results in: \(b = 26/13\).

Key Concepts

Rearranging Equations

To start with solving linear equations, it's crucial to rearrange terms properly. Rearranging equations means altering the position of terms to simplify solving them. In our example, the equation given is initially unbalanced with an unknown mixed on both sides:

• The original equation is: \(4b = 26 - 9b\)
• To rearrange it, we need to move the \(9b\) term to the left side. This is done by adding \(9b\) to both sides: \(4b + 9b = 26\)

By doing this, we isolate all the terms involving the variable on one side, which simplifies solving the equation. Remember, whatever you do to one side of the equation, you must do to the other to maintain equality. This is a fundamental rule in solving equations.

Combining Like Terms

After rearranging the equation, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In the equation \(4b + 9b = 26\), both terms on the left include the variable \(b\). This means:

• Add the coefficients (the numbers in front of the variables) together:
• \(4 + 9 = 13\)

So, \(4b + 9b\) simplifies to \(13b\). Combining like terms simplifies the problem and helps in progressing towards isolating the variable.
Once the equation is simplified to \(13b = 26\), the next steps become much clearer and straightforward.

Solving for a Variable

The final stage in solving the equation is to solve for the variable. This means isolating the variable to find its value.

• Starting with the simplified equation: \(13b = 26\).
• We need to divide both sides by \(13\), the coefficient of \(b\):
• \(b = \frac{26}{13}\)
• Simplifying the right side gives us \(b = 2\).

By carrying out this division, we find that \(b\) is equal to 2. This value of \(b\) is the solution to the original equation. Solving for a variable might seem challenging at first, but by following the necessary steps, it becomes a straightforward process. Always double-check your operations to ensure accuracy.