Solving Systems Of Equations Strategically Variable Selection For Substitution

Sarita faces a classic algebra problem a system of two linear equations with two variables She needs to decide the most efficient way to solve it and substitution seems like the best route This article will delve into the strategic considerations for solving systems of equations using substitution guiding Sarita and anyone else facing similar challenges to make the optimal choice for variable selection and streamline the solution process

Understanding the Problem The Foundation of Success

Before diving into the solution it is crucial to understand the given equations

2x + 3y = 25
4x + 2y = 22

This system represents two linear relationships between x and y Our goal is to find the values of x and y that satisfy both equations simultaneously The substitution method involves solving one equation for one variable and then substituting that expression into the other equation This eliminates one variable allowing us to solve for the remaining one

To choose the best variable to solve for we should look for a variable with a coefficient of 1 or -1 in either equation If such a variable exists it simplifies the process of isolating it This avoids fractions and makes the subsequent substitution and simplification steps easier However in this case we don't have a coefficient of 1 or -1 So we need to think about the next best option which is the smallest coefficient In the first equation 2x + 3y = 25 the coefficients are 2 and 3 In the second equation 4x + 2y = 22 the coefficients are 4 and 2 The smallest coefficient overall is 2 which appears twice once with x in the first equation and once with y in the second equation This means we have two potential candidates for easy solving Either we solve the first equation for x or we solve the second equation for y Let's examine both scenarios to decide the best course of action

If we solve the first equation 2x + 3y = 25 for x we would subtract 3y from both sides resulting in 2x = 25 - 3y Then we would divide both sides by 2 giving us x = (25 - 3y) / 2 While this is a valid step it introduces a fraction which can make the following steps more cumbersome Similarly if we solve the second equation 4x + 2y = 22 for x we would subtract 2y from both sides resulting in 4x = 22 - 2y Then we would divide both sides by 4 giving us x = (22 - 2y) / 4 This also involves a fraction although it can be simplified to x = (11 - y) / 2

Now let's consider solving for y In the first equation 2x + 3y = 25 if we solve for y we would subtract 2x from both sides resulting in 3y = 25 - 2x Then we would divide both sides by 3 giving us y = (25 - 2x) / 3 Again this introduces a fraction However if we solve the second equation 4x + 2y = 22 for y we would subtract 4x from both sides resulting in 2y = 22 - 4x Then we would divide both sides by 2 giving us y = 11 - 2x This is the most promising option because it avoids fractions entirely The expression for y is a simple linear expression in terms of x

Therefore the best strategy is to solve the second equation for y This will lead to a cleaner substitution and ultimately a simpler solution process By carefully considering the coefficients and the resulting expressions we can make an informed decision that saves time and effort This strategic approach is a key element of mathematical problem-solving

Strategic Variable Selection A Deep Dive

When tackling systems of equations with substitution the initial choice of which variable to isolate can significantly impact the complexity of the solution process The goal is to select a variable that minimizes the introduction of fractions or complex expressions leading to a more streamlined and efficient solution In Sarita's case we have the following system

2x + 3y = 25
4x + 2y = 22

As discussed earlier the ideal scenario is to find a variable with a coefficient of 1 or -1 If such a variable exists isolating it is straightforward and avoids fractions However in this system none of the variables have a coefficient of 1 or -1 This means we need to employ a different strategy to determine the best variable to solve for

The next best approach is to look for the smallest coefficient This is because dividing by a smaller number is less likely to result in unwieldy fractions or decimals In this system the smallest coefficient is 2 which appears twice once with x in the first equation and once with y in the second equation This gives us two potential paths to explore solving the first equation for x or solving the second equation for y

Let's analyze the consequences of each choice If we solve the first equation 2x + 3y = 25 for x we would perform the following steps

1 Subtract 3y from both sides 2x = 25 - 3y 2 Divide both sides by 2 x = (25 - 3y) / 2

This results in an expression for x that involves a fraction which could make subsequent substitutions and simplifications more complex

Now let's consider solving the second equation 4x + 2y = 22 for y

1 Subtract 4x from both sides 2y = 22 - 4x 2 Divide both sides by 2 y = 11 - 2x

In this case solving for y results in a simple linear expression without any fractions This is a significant advantage as it simplifies the substitution process and reduces the chances of making errors

By comparing the two scenarios it becomes clear that solving the second equation for y is the more strategic choice It leads to a cleaner expression and minimizes the complexity of the subsequent steps This decision-making process is a crucial aspect of problem-solving in algebra It's not just about finding the answer but also about finding the most efficient path to the solution By carefully analyzing the equations and considering the potential outcomes of each choice we can make informed decisions that save time and effort

Furthermore recognizing patterns and developing intuition about which variables are likely to lead to simpler solutions is a valuable skill in mathematics This strategic approach extends beyond this specific problem and can be applied to a wide range of algebraic challenges It emphasizes the importance of not just blindly following a method but also understanding the underlying principles and making informed choices based on the specific characteristics of the problem

Step-by-Step Solution Streamlining the Process

Having strategically chosen to solve the second equation for y we can now proceed with the substitution method Step-by-step solution not only makes the problem more approachable but also solidifies understanding of the technique

1 We have already solved the second equation for y

y = 11 - 2x

This is our expression for y in terms of x

2 Substitute this expression for y into the first equation

The first equation is 2x + 3y = 25 Replacing y with (11 - 2x) gives us

2x + 3(11 - 2x) = 25

This step eliminates y from the equation leaving us with an equation in just one variable x

3 Simplify and solve for x

Distribute the 3 2x + 33 - 6x = 25

Combine like terms -4x + 33 = 25

Subtract 33 from both sides -4x = -8

Divide both sides by -4 x = 2

We have now found the value of x

4 Substitute the value of x back into the expression for y

We have y = 11 - 2x Substituting x = 2 gives us

y = 11 - 2(2)

y = 11 - 4

y = 7

Now we have the value of y

5 Verify the solution

To ensure our solution is correct we should substitute the values of x and y back into both original equations

First equation 2x + 3y = 25

2(2) + 3(7) = 4 + 21 = 25

The first equation is satisfied

Second equation 4x + 2y = 22

4(2) + 2(7) = 8 + 14 = 22

The second equation is also satisfied

Therefore our solution is correct

By following this step-by-step process we have successfully solved the system of equations using substitution The solution is x = 2 and y = 7 This systematic approach not only helps in solving the problem accurately but also reinforces the understanding of the underlying principles and techniques Each step builds upon the previous one creating a clear and logical path to the solution This is a valuable skill in mathematics where breaking down complex problems into smaller manageable steps is often the key to success

Conclusion Sarita's Optimal Choice

In conclusion Sarita should strategically choose to solve for y in the second equation (4x + 2y = 22) This approach avoids fractions in the initial substitution step leading to a more straightforward solution By carefully considering the coefficients and the resulting expressions Sarita can efficiently solve the system of equations using substitution Remember strategic variable selection is a crucial skill in algebra that can simplify complex problems