Question

Your school is building a new computer center. Four hundred square feet of the center will be available for computer stations. Each station requires 20 square feet. You want to find how many computer stations can be placed in the new center. You write the equation \(20 x=400\) to model the situation. What do \(20, x,\) and 400 represent? Solve the equation. Check your solution.

Short answer

The equation provided, \(20x = 400\), was used to model the problem, where '20' represents the square footage needed by each computer station, 'x' is the number of stations, and '400' is the total space available. Solving the equation yields 'x' = 20, confirming that it is possible to fit 20 computer stations into the 400 square foot space.

Step-by-step solution

Understanding the equation

In the equation \(20x = 400\), 20 represents the square footage needed for each computer station, x represents the count of stations we need to find, and 400 is the total square footage available for stations.

Solve for 'x'

In order to find the value of 'x', which represents the number of computer stations that can fit into the space, we need to rearrange the equation and solve for 'x'. Thus, we divide both sides of the equation by 20. The operation will look like this: \(x = 400\div 20\).

Compute the value of 'x'

Computing the result gives: \(x = 20\). This means that 20 computer stations can be accommodated in the computer center.

Check the solution

Plug the value of 'x' back into the original equation as a check: \(20 * 20 = 400\). This results in the equation: \(400 = 400\), affirming our solution is correct. We can therefore conclude that our solution is accurate.

Key concepts

Algebraic Problem Solving

Algebraic problem solving is like a puzzle, where you need to find the missing pieces to complete the picture. In the context of our example, we are given a real-world scenario—a new computer center needing computer stations.

The first piece of this puzzle is to identify what we know: the total area (400 square feet) and how much space one station occupies (20 square feet). Algebra comes into play as we translate this situation into an equation, in this case, a linear equation:

퐒퐓퐑퐀퐓퐄퐆퐈퐂퐀퐋퐋퐘 퐃퐄퐀퐋 퐖퐈퐓퐇퐄퐀퐂퐇퐒퐄퐓 퐎퐅퐍퐔퐌퐁퐄퐑퐒퐕퐀퐑퐈퐀퐁퐋퐄 ```퐒trategically,퐏roble퐚nd퐒olving```eals with each퐒et of numbers and퐒ariables in an equation퐨r퐢n algebraic퐂oncept 퐋ike the puzzle, to make sense of it and solve it. The success of problem solving depends on a clear understanding of what each part of the equation represents and how it ties back to the problem at hand.

Linear Equation Model

The linear equation model is an incredibly useful mathematical tool, especially when dealing with problems involving constant rates or proportions.

In the computer center example, the linear model is the equation: ter more text here ter more text here ter more text here ter more text here ter more text here 기퐰hich simplifies to 퐱 = 퐭otal area 퐛y 퐚rea per station. It's called linear because, graphically, if we were to plot the number of stations against the area needed, it would form a straight line. The constants in our equation, namely 20 and 400, give us a fixed rate, making the equation predictable and, therefore, solvable.

By understanding the linear model, we can better appreciate the relationship between the variable (the number of stations) and constants, which can help in finding practical solutions.

Equation Solving Steps

Breaking down equation solving into clear, manageable steps is essential for understanding and mastery. Let's revisit the given problem for an overview of these steps.

First, Step 1: Understand the equation. This is about contextualizing what each term represents in real life. For our example, '20' is the square footage each station takes, 'x' is the number of stations, and '400' is the total available square footage.

Moving on to Step 2: Solve for 'x', or our unknown variable. This typically involves isolating 'x' on one side of the equation. Here, dividing both sides by 20 achieves this goal, leading us to Step 3.

In Step 3: Compute the value of 'x', simple arithmetic gives us 'x = 20'. This is the practical answer to our real-world problem – the number of computer stations we can fit into the center.

Lastly, Step 4: Check the solution reassures us that our answer makes logical sense when plugged back into the original equation.

Through these steps, problem solvers can systematically address and solve equations with confidence and precision.