Solving A System Of Linear Equations: Step-by-Step
Hey guys! Today, we're diving into a super common math problem: solving a system of linear equations. Specifically, we're tackling this one: 2x + 5y = 16 and 3x + 2y = 2. We need to figure out which of the following options is the correct solution: a) (2, 2) b) (3, 1) c) (4, 0) d) (1, 3) e) (0, 3).
Understanding Systems of Linear Equations
Before we jump into solving, let's quickly recap what a system of linear equations actually is. A system of linear equations is simply a set of two or more linear equations that we're trying to solve simultaneously. Each equation represents a straight line, and the solution to the system is the point (or points) where these lines intersect. This point satisfies all equations in the system. Solving these systems is a fundamental skill in algebra and has tons of applications in various fields, like economics, engineering, and computer science. Think about scenarios where you have multiple constraints or conditions, and you need to find a single set of values that satisfies all of them. That's where solving systems of equations comes in handy!
There are several methods to solve these systems, including substitution, elimination (also known as addition), and graphing. We'll primarily focus on the elimination method here because it’s often quite efficient for this type of problem. Each method has its strengths and weaknesses, and the best choice often depends on the specific equations you're dealing with. Substitution is great when one equation is already solved for one variable. Elimination shines when the coefficients of one variable are easily made opposites. And graphing is a visual approach that can be helpful for understanding the concept, though it's not always the most precise for finding exact solutions.
When you encounter these problems, remember to check your solutions by plugging them back into the original equations. This ensures that you haven't made any algebraic errors along the way. It’s a simple step that can save you from choosing the wrong answer, especially in a test setting! Also, keep an eye out for special cases like parallel lines (no solution) or coinciding lines (infinite solutions). These situations can add a bit of a twist to the problem, but with practice, you'll become adept at recognizing them. Systems of linear equations are like puzzles, and each method is a tool to help you piece together the solution. So, let's get to it and crack this one!
Method 1: Elimination Method
The elimination method involves manipulating the equations so that the coefficients of one of the variables are opposites. Then, we add the equations together, eliminating that variable and allowing us to solve for the remaining one. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.
Let's apply this to our system:
2x + 5y = 16 3x + 2y = 2
Our goal is to make either the 'x' coefficients or the 'y' coefficients opposites. Let's choose to eliminate 'x'. To do this, we need to find the least common multiple (LCM) of 2 and 3, which is 6. We'll multiply the first equation by 3 and the second equation by -2 to achieve coefficients of 6 and -6 for 'x'.
Multiply the first equation by 3:
3 * (2x + 5y) = 3 * 16 6x + 15y = 48
Multiply the second equation by -2:
-2 * (3x + 2y) = -2 * 2 -6x - 4y = -4
Now, we have the modified system:
6x + 15y = 48 -6x - 4y = -4
Next, add the two equations together:
(6x + 15y) + (-6x - 4y) = 48 + (-4) 6x - 6x + 15y - 4y = 44 11y = 44
Now, solve for 'y':
y = 44 / 11 y = 4
Great! We've found that y = 4. Now, substitute this value back into one of the original equations to solve for 'x'. Let's use the first original equation:
2x + 5y = 16 2x + 5(4) = 16 2x + 20 = 16 2x = 16 - 20 2x = -4 x = -4 / 2 x = -2
So, we have x = -2 and y = 4. Therefore, the solution to the system of equations is (-2, 4). However, this solution isn't among the choices provided. Let's double-check our work to make sure we didn't make any arithmetic errors.
Looking back, it seems we made a mistake by assuming the correct answer was among the given choices. We should always verify our solutions, but let’s re-examine the original problem and the steps we took.
Retracing the steps:
First Equation: 2x + 5y = 16 Second Equation: 3x + 2y = 2
Multiply the first equation by 3: 6x + 15y = 48 Multiply the second equation by -2: -6x - 4y = -4
Adding them gives: 11y = 44 So, y = 4
Substituting y = 4 into the first equation: 2x + 5(4) = 16 2x + 20 = 16 2x = -4 x = -2
The correct solution is indeed (-2, 4), which is not among the options provided. This implies there might be an error in the given alternatives. Since we are forced to choose among the provided options, let’s test each one to see which comes closest.
Method 2: Testing Each Option
Since we didn't find our solution in the list, let's test each option provided to see which one satisfies the equations or comes closest to satisfying them.
a) (2, 2)
- 2x + 5y = 2(2) + 5(2) = 4 + 10 = 14 (Not 16)
- 3x + 2y = 3(2) + 2(2) = 6 + 4 = 10 (Not 2)
b) (3, 1)
- 2x + 5y = 2(3) + 5(1) = 6 + 5 = 11 (Not 16)
- 3x + 2y = 3(3) + 2(1) = 9 + 2 = 11 (Not 2)
c) (4, 0)
- 2x + 5y = 2(4) + 5(0) = 8 + 0 = 8 (Not 16)
- 3x + 2y = 3(4) + 2(0) = 12 + 0 = 12 (Not 2)
d) (1, 3)
- 2x + 5y = 2(1) + 5(3) = 2 + 15 = 17 (Close to 16)
- 3x + 2y = 3(1) + 2(3) = 3 + 6 = 9 (Not 2)
e) (0, 3)
- 2x + 5y = 2(0) + 5(3) = 0 + 15 = 15 (Close to 16)
- 3x + 2y = 3(0) + 2(3) = 0 + 6 = 6 (Not 2)
Looking at these results, none of the options perfectly satisfy both equations. However, option d) (1, 3) comes closest to satisfying the first equation (2x + 5y = 16), resulting in 17, which is just off by 1. Although it doesn't satisfy the second equation well, it's the closest we can get from the provided choices. Therefore, considering the constraints and the likely presence of an error in the options, we choose option d) as the 'best' answer.
Conclusion
In conclusion, while the accurate solution to the system of equations 2x + 5y = 16 and 3x + 2y = 2 is (-2, 4), which was obtained using the elimination method, this option was not available among the provided choices. After testing each of the given alternatives, option d) (1, 3) provided the closest result to satisfying both equations. Therefore, under the given circumstances, option d) is the most reasonable answer. Always remember to double-check your work and the provided options to ensure accuracy. Math can be tricky sometimes, but with a bit of patience, you can always find the solution! Keep practicing, and you'll nail these problems every time! Remember, guys, math is like a muscle; the more you use it, the stronger it gets!