Problem 21 - Prealgebra

Question

Solve each equation. Check your solution. $$3 u=51$$

Step-by-Step Solution

Answer

The solution is \(u = 17\).

Step 1: Identify the Goal

We are solving the equation \(3u = 51\). Our goal is to find the value of \(u\) that makes this equation true.

Step 2: Isolate the Variable

In order to find \(u\), we need to get \(u\) by itself on one side of the equation. Currently, \(u\) is being multiplied by 3. To isolate \(u\), we will divide both sides of the equation by 3.

Step 3: Perform the Division

Divide both sides of the equation by 3 to solve for \(u\):\[ u = \frac{51}{3} \]. Simplifying the division gives us \( u = 17 \).

Step 4: Check the Solution

To ensure our solution is correct, we substitute \(u = 17\) back into the original equation. The equation is \(3u = 51\). Substituting gives: \(3 \times 17 = 51\). Calculating further: \(51 = 51\). Since both sides are equal, our solution is verified.

Key Concepts

Solving Equations

Finding the solution to an equation is like solving a mystery. We start with an equation such as \(3u = 51\). The main aim is to figure out what value of \(u\) makes this equation true. In terms of linear equations, this involves manipulating the equation to get the variable, \(u\), by itself on one side. In our example, \(u\) is multiplied by 3. To "undo" this multiplication, we divide both sides by 3. This is because dividing by 3 cancels out the multiplication on \(u\). This process is a step-by-step path towards finding that \(u = 17\). Each step should be done carefully to ensure accuracy, which is crucial in solving any kind of equation.

Prealgebra

Prealgebra is all about getting comfortable with numbers and how they behave in equations. It lays the foundation for all future math studies, featuring elements such as basic operations, understanding variables, and arithmetic with numbers. One critical skill in prealgebra is recognizing operations that "undo" each other, like division and multiplication. For example, in the original equation \(3u = 51\), the number 3 is multiplying \(u\). Division is used to "get rid of" this multiplier. We divide both sides by 3 to reverse the multiplication, making it a simpler and more straightforward task to isolate \(u\). Knowing these operations can simplify problems and unlock solutions.

Checking Solutions

After you've solved an equation, it's important to check your solution to ensure it's correct. This verification step acts like checking your work to make sure nothing went wrong in the calculations. With our equation \(3u = 51\), after calculating \(u = 17\), we substitute \(17\) back into the original equation to see if it satisfies the equation. We replace \(u\) in \(3u\) with 17, giving us \(3 \times 17 = 51\). This reduces to \(51 = 51\), confirming that both sides match and the solution is indeed correct. Checking solutions is a reassuring step that reinforces your understanding and confidence in solving equations.