Understanding and solving systems of linear equations is a fundamental skill in the world of mathematics. These systems can be powerful tools for modeling and solving real-world problems. In this blog post, we'll explore the process of setting up a system of two linear equations. Specifically, we'll focus on situations where equations involve rates, quantities, and units. By the end, you'll be equipped with the knowledge to tackle problems that require setting up and solving systems of equations.
**Quick Hint - If you ever have two unknowns, just create two equations and you can solve your problem if it has a solution**
1. Identify the Variables (What are your unknowns):
Before diving into the problem, it's crucial to identify the variables that represent the unknowns in your system. For instance, if you are dealing with a scenario involving rates and quantities, the variable will usually represent either the unknown quantities of two items or the unknown rates at which those two items are changing (e.g. price of an item). This step sets the stage for translating the problem into mathematical equations.
2. Create your Equations:
Let's consider an example. Suppose you're dealing with a situation involving a total value. Pay attention to units and rates in the problem. If you have two total costs for two different combination of quantities when purchasing two items, we can set up the equation as follows:
Let b represent the price (rate) in dollars per pound for buying a certain quantity of bananas and let a represent the price (rate) in dollars per pound for buying a certain quantity of apples.
If you paid $4.50 when buying 3 pounds of bananas and 5 pounds of apples, and your friend paid $7.35 when buying 5 pounds of bananas and 8 pounds of apples, then our two equations would be set up as follows:
3b+5a=4.5
5b+8a=7.35
3. Consistent Units:
Ensure that the units are consistent throughout the problem. If the rate is given in dollars per pounds and the quantity is in pounds, the resulting cost will be in dollars. This consistency is vital for the correctness of your equations.
4. Use Descriptive Variables:
Choose variables that make sense in the context of the problem. If you're dealing with quantities of items, use variables that represent those items. This makes it easier to interpret the solutions once you've solved the system. In our systems we used b to represent the price of the bananas per pound and a to represent the price of the apples per pound.
5. Solving the System:
Once you have your system of equations, you can solve them using various methods such as substitution, elimination, or matrices. The solution to the system will provide the values of the variables that satisfy both equations simultaneously. When using systems to solve problems in the SAT exam, you can always graph both equation (just change your two variables to x and y for graphing purposes) and find the solution by checking the point the two lines share in common aka. point of intersection.
Conclusion:
Setting up a system of two linear equations is a valuable skill that finds applications in various fields, from physics to economics. By paying attention to units, rates, and quantities, you can effectively translate real-world problems into mathematical equations. This lays the groundwork for finding solutions that make sense in the given context. As you practice and encounter different scenarios, you'll become more adept at modeling diverse situations with systems of linear equations.
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