Solving 2x + 3y = 4 And 3x - 2y = 13 Using Substitution And Elimination

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Introduction

Hey guys! Today, we're diving into a fundamental topic in mathematics: solving systems of linear equations. We'll be tackling a specific problem: finding the values of x and y that satisfy the following two equations simultaneously:

  1. 2x + 3y = 4
  2. 3x - 2y = 13

We'll explore two powerful methods for solving such systems: substitution and elimination. Both methods are crucial tools in algebra and have wide applications in various fields like engineering, economics, and computer science. Mastering these techniques will not only help you ace your math exams but also equip you with valuable problem-solving skills for real-world scenarios. So, let's buckle up and get started!

Understanding Systems of Linear Equations

Before we jump into the solution, let's take a moment to understand what a system of linear equations actually represents. In simple terms, it's a set of two or more linear equations that share the same variables. A linear equation, as the name suggests, represents a straight line when plotted on a graph. When we have a system of linear equations, we're essentially looking for the point(s) where these lines intersect. The coordinates of this intersection point represent the solution that satisfies all the equations in the system.

In our case, we have two equations, each with two variables (x and y). This means each equation represents a line in a two-dimensional plane. The solution to the system will be the point (x, y) where these two lines intersect. There are three possibilities for the intersection:

  1. One unique solution: The lines intersect at a single point.
  2. No solution: The lines are parallel and never intersect.
  3. Infinitely many solutions: The lines are the same, overlapping each other.

Our goal is to determine which of these scenarios applies to our system and, if there's a unique solution, to find the values of x and y.

Method 1: Substitution

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This process eliminates one variable, leaving us with a single equation in one variable, which we can easily solve. Let's apply this method to our system:

  1. 2x + 3y = 4
  2. 3x - 2y = 13

Step 1: Solve one equation for one variable.

Let's solve the first equation for x:2x = 4 - 3yx = (4 - 3y) / 2

Step 2: Substitute the expression into the other equation.

Now, substitute this expression for x into the second equation:3 * ((4 - 3y) / 2) - 2y = 13

Step 3: Solve the resulting equation for the remaining variable.

Let's simplify and solve for y: (12 - 9y) / 2 - 2y = 13 12 - 9y - 4y = 26 -13y = 14 y = -14 / 13

Step 4: Substitute the value back to find the other variable.

Now that we have the value of y, we can substitute it back into either of the original equations or the expression we derived for x. Let's use the expression for x:x = (4 - 3*(-14/13)) / 2 x = (4 + 42/13) / 2 x = (52/13 + 42/13) / 2 x = (94/13) / 2 x = 47 / 13

Therefore, using the substitution method, we find that x = 47/13 and y = -14/13. Now, let's verify this solution by plugging these values back into the original equations. This is a crucial step to ensure we haven't made any errors in our calculations. Substituting into the first equation: 2 * (47/13) + 3 * (-14/13) = 94/13 - 42/13 = 52/13 = 4

And substituting into the second equation: 3 * (47/13) - 2 * (-14/13) = 141/13 + 28/13 = 169/13 = 13

Both equations hold true, so our solution is correct. The substitution method provides a systematic way to solve systems of equations by reducing the problem to a single equation with one variable.

Method 2: Elimination

The elimination method involves manipulating the equations so that the coefficients of one variable are opposites. Then, by adding the equations together, that variable is eliminated, leaving us with a single equation in one variable. Let's apply this method to our system:

  1. 2x + 3y = 4
  2. 3x - 2y = 13

Step 1: Multiply the equations to make the coefficients of one variable opposites.

To eliminate y, we can multiply the first equation by 2 and the second equation by 3: (2x + 3y) * 2 = 4 * 2 => 4x + 6y = 8 (3x - 2y) * 3 = 13 * 3 => 9x - 6y = 39

Step 2: Add the equations together to eliminate one variable.

Now, add the two equations: (4x + 6y) + (9x - 6y) = 8 + 39 13x = 47 x = 47 / 13

Step 3: Substitute the value back to find the other variable.

Now that we have the value of x, we can substitute it back into either of the original equations. Let's use the first equation:2 * (47/13) + 3y = 4 94/13 + 3y = 4 3y = 4 - 94/13 3y = 52/13 - 94/13 3y = -42/13 y = -14/13

Therefore, using the elimination method, we also find that x = 47/13 and y = -14/13. We obtained the same solution as with the substitution method, which reinforces the correctness of our answer. The elimination method is particularly useful when the coefficients of one variable are easily made opposites by multiplication. It offers another systematic approach to solving systems of equations.

Comparing Substitution and Elimination

Both substitution and elimination are powerful methods for solving systems of linear equations, but they have their own strengths and weaknesses. The best method to use often depends on the specific system you're dealing with.

  • Substitution: This method is generally easier to use when one of the equations is already solved for one variable or can be easily solved. It's also a good choice when you have a system with variables that have coefficients of 1. The key is to isolate one variable in one equation and then substitute that expression into the other equation. This reduces the system to a single equation with one variable, which is straightforward to solve.

  • Elimination: This method shines when the coefficients of one of the variables are already opposites or can be easily made opposites by multiplying one or both equations by a constant. It's particularly effective when you have equations in standard form (Ax + By = C) and the coefficients of one variable are multiples of each other. By adding or subtracting the equations, you can eliminate one variable and solve for the other.

In our example, both methods worked effectively, yielding the same solution. However, in some cases, one method might be more efficient than the other. For instance, if one equation was already solved for x or y, substitution would likely be the quicker route. Conversely, if the coefficients of either x or y were easily made opposites, elimination might be the preferred approach.

Ultimately, the choice of method is a matter of personal preference and the specific characteristics of the system of equations. The more you practice with both methods, the better you'll become at recognizing which one will be most efficient in a given situation.

Conclusion

Alright guys, we've successfully solved the system of equations 2x + 3y = 4 and 3x - 2y = 13 using both substitution and elimination methods. We found that x = 47/13 and y = -14/13 is the solution that satisfies both equations. We also discussed the advantages and disadvantages of each method, giving you a better understanding of when to use each one.

Solving systems of equations is a fundamental skill in mathematics, and mastering these methods will open doors to more advanced topics. Keep practicing, and you'll become a pro at solving these types of problems in no time! Remember, the key is to understand the underlying concepts and choose the method that best suits the given system. With practice and perseverance, you'll be able to tackle any system of equations that comes your way. So, keep up the great work, and happy problem-solving!