Problem 30 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Use division to solve the equation. $$ 288=16 u $$

Step-by-Step Solution

Answer

The solution to the equation 288=16u, solved by division, is u=18.

Step 1: Identify the problem

The problem presents an equation 288=16u that we need to solve for the unknown 'u'. In order to isolate 'u', we need to use the inverse of multiplication, which is division.

Step 2: Divide both sides of the equation by 16

To isolate 'u', divide both sides of the equation by 16. This will look like: \( \frac{288}{16} = \frac{16u}{16} \). The division of 16u by 16 cancels out, leaving 'u' on the right side of the equation.

Step 3: Calculate the result

Now calculate the left side: \( \frac{288}{16} = 18 \), so the solution is \(18 = u\)

Key Concepts

Division

Division is an essential math operation, and it's particularly useful when solving equations. In the given exercise, we are looking at the division of both sides of an equation by a constant number. The goal here is to simplify the equation, so the variable, 'u' in this case, is isolated and can be solved for. When you divide both sides by the same number, you maintain the equation's balance. It’s like cutting a cake into equal parts; everyone still gets the same amount, ensuring fairness. Here’s why division is useful:

• It helps reduce multiplicative expressions.
• Allows simplification that brings clarity.
• Helps in easily finding one unknown in a balanced equation.

Whenever you see an equation like our example, where a variable is multiplied by a number, think 'division' to break it down.

Inverse Operations

Understanding inverse operations is like knowing the opposite sides of a coin. In math, every operation has an inverse that can undo it. For multiplication, the inverse operation is division. This concept is fundamental when solving equations as it helps in reversing the effect of an operation to isolate a variable. In the equation 288 = 16u, multiplication is used to relate '16u' and '288.' To solve for 'u,' we use division, which is an inverse operation of multiplication. By dividing both sides by 16, you're effectively canceling out the multiplication, making 'u' stand alone. Here are key aspects of inverse operations in solving equations:

• They maintain the equality of the equation while simplifying it.
• Help in transitioning back to the original value of a variable.
• Are fundamental in checking the correctness of solutions.

Think of inverse operations as the "undo" button for mathematical procedures.

Isolating Variables

Isolating variables is a critical step when solving equations, paving the way to find the exact value that satisfies the equation. The term "isolating" might sound complex, but it simply means to get the variable by itself on one side of the equation, away from numbers or other variables. In our example, the variable we aim to isolate is 'u.' Once we divide both sides by 16, the right side of the equation simplifies to just 'u,' effectively isolating it. This process allows us to directly see that 'u' equals the result obtained from the division of 288 by 16. Why is isolating variables so crucial?

• It simplifies the process of solving equations.
• Enables clear understanding and interpretation of the variable's value.
• Makes it easier to verify and check solutions by looking at them more transparently.

The skill of isolating a variable becomes incredibly handy in algebra and beyond, simplifying more complex equations with multiple steps.