Problem 89 - College Algebra

Question

solve each equation. $$ \left[(3+6)^{2}+3\right] \cdot 4=-54 x $$

Step-by-Step Solution

Answer

The solution to the equation is \(x=-\frac{56}{9}\).

Step 1: Simplify the equation

Start by simplifying the expression inside the brackets. Hence, \[ \left[(3+6)^{2}+3\right] \cdot 4=-54x \] is equivalent to \[ (9^{2}+3) \cdot 4=-54x \] resulting in \[ (81+3) \cdot 4=-54x \] which simplifies to \[ 84 \cdot 4=-54x \]

Step 2: Perform scalar multiplication

Next, perform the multiplication operation, \[ 84 \cdot 4=-54x \] simplifies to \[ 336=-54x\]

Step 3: Solve for x

In this step, divide both sides by -54 to solve for \(x\), \[ \frac{336}{-54} = x\] which simplifies to \[ x=-\frac{56}{9}\].

Key Concepts

Simplifying Expressions

Simplifying expressions is one of the first steps in solving complex equations. It involves breaking down expressions into their simplest form, which helps make calculations easier. In the given problem, we start with the expression inside the brackets: \((3+6)^2 + 3\). To simplify it, follow these steps:

• First, solve the expression inside the parentheses: \(3 + 6 = 9\).
• Next, square the result: \(9^2 = 81\).
• Then, add 3 to the squared result: \(81 + 3 = 84\).

Now, the complex expression \((3 + 6)^2 + 3\) is simplified to 84, making it easier to use in further calculations. Identifying and simplifying parts of an equation one step at a time is crucial to solve any equation efficiently.

Scalar Multiplication

Scalar multiplication is the process of multiplying a scalar, which is just a real number, by an expression or number. In the context of the given equation, after simplification, we have the expression \(84 \cdot 4 = -54x\). Here's how you handle scalar multiplication:

• Multiply the simplified expression 84 by the scalar 4: \(84 \times 4\).
• The calculation gives 336, rewritten as the equation: \(336 = -54x\).

Scalar multiplication is a straightforward operation, but it is essential in combining terms correctly, which can simplify solving equations. Always ensure you correctly multiply numbers to avoid errors in later steps.

Division in Equations

Once you have simplified the equation and performed necessary multiplications, the next step is solving for the unknown variable using division. For our equation \(336 = -54x\), we need to isolate \(x\):

• To solve for \(x\), divide both sides of the equation by \(-54\): \(\frac{336}{-54} = x\).
• This division gives \(x = -\frac{56}{9}\).

Division is essential for isolating the variable in an equation and simplifying it to find the value of unknowns. Make sure to apply the division operation on both sides of the equation to maintain equality.