Question

You are playing a new computer game. For every eight screens you complete, you receive a bonus. You want to know how many bonuses you will receive after completing 96 screens. You write the equation \(8 x=96\) to model the situation. What do \(8, x,\) and 96 represent? Solve the equation. Check your solution.

Short answer

In the context of the game, '8' represents the number of screens completed to get a bonus, 'x' represents the number of bonuses, and '96' is the total number of screens completed. Solving the equation \(8x = 96\) gives \(x = 12\), which means 12 bonuses are received after completing 96 screens.

Step-by-step solution

Interpretation

Let's interpret the elements, where '8' represents the number of completed screens that yield a bonus in the game. The variable 'x' stands for the total number of bonuses, and '96' symbolizes the overall completed screens.

Solving the Equation

To resolve the equation \(8x=96\), you need to isolate 'x'. You can do this by dividing both sides of the equation by 8. This gives \(x = \frac{96}{8}\).

Checking the Solution

To ensure the accuracy of the found value for 'x', substitute 'x' in the original equation with this value and check if the equation holds true. Therefore, substituting back into the equation gives \(8 * 12 =96\). The equation is correct, so the solution for 'x' is confirmed to be accurate.

Key concepts

Linear Equation Examples

When dealing with mathematics, the concept of a linear equation is a fundamental building block that applies to various real-world scenarios, just like in the new computer game where you calculate bonuses. A classic example of a linear equation is the one provided in the exercise: \(8x = 96\). Here, the equation models a situation where completing a certain number of tasks (in this case, screens) results in a reward (bonus).

For linear equation examples, imagine scenarios such as calculating distances traveled over time at a constant speed, represented by \(d = rt\), where \(d\) stands for distance, \(r\) for rate, and \(t\) for time. Another one might involve budgeting, with an equation stating a relationship such as \(y = mx + b\), where \(y\) represents the total cost, \(m\) is the cost per item, \(x\) is the number of items, and \(b\) is the base cost. These equations allow us to predict outcomes and make informed decisions based on provided relationships.

Equation Solving Steps

To solve an equation effectively, it's essential to follow systematic steps that lead to isolating the variable of interest. In the game-related example, we're keen to find how many bonuses you'd get, which means we need to find the value of \(x\).

1. Understand the Equation: Begin by identifying what each term represents, as was done with the '8' screens, the '96' completed screens, and 'x' being the number of bonuses.
2. Isolate the Variable: Use algebraic operations to make 'x' the subject of the equation. You might need to add, subtract, multiply, divide, or use a combination of these operations depending on the initial form of the equation. In our example, we divide both sides by '8'.
3. Perform the Operation: Calculate the right side of the equation after the algebraic operation. You'll get a numerical value that represents the variable's possible solution, yielding \(x = 12\).
4. Check Your Solution: Always verify your answer by substituting the variable back into the original equation to see if it balances. If it does, as with \(8 * 12 = 96\), your solution is correct.

By following these steps meticulously, you increase your chances of accurately solving linear equations and understanding the mechanisms behind them.

Variable Isolation

Variable isolation is a critical skill in algebra, allowing one to find the value of an unknown in an equation. The goal is to get the variable alone on one side of the equation - essentially so that the equation reads 'Variable = (number)' for clarity and simplicity.

To do this, we perform operations that 'undo' whatever is being done to the variable. In the equation \(8x = 96\), 'x' is being multiplied by '8'. To isolate 'x', we must do the opposite of multiplying by '8', which is dividing by '8'. When we divide both sides of the equation by '8', we get \(x = \frac{96}{8}\), which simplifies to 12 after doing the calculation.

Principles of Variable Isolation

• Inverse Operations: Use operations opposite to those applied to the variable.
• Balancing the Equation: Perform the same operations on both sides to maintain the equation's balance.
• Clearing Fractions or Decimals: Whenever possible, multiply by common denominators to rid the equation of fractions or decimals before isolating the variable.
• Checking the Result: Always substitute the isolated variable back into the original equation to ensure the correctness of your solution.

With practice, variable isolation becomes an intuitive process that makes solving equations much more accessible and less intimidating.