Solving 5x-2y=15 And 2x+4y=12 The Best First Step With Substitution

by ADMIN 68 views

When faced with a system of equations, the substitution method offers a powerful approach to finding solutions. The question at hand presents us with a specific system:

5x - 2y = 15
2x + 4y = 12

Our goal is to determine the most efficient initial step in solving this system using substitution. The options given are:

  • A. Solve for x in the first equation.
  • B. Solve for y in the first equation.
  • C. Solve for x in the second equation.
  • D. Solve for y in the second equation.

To make an informed decision, let's delve into the strategy behind substitution and evaluate each option.

Understanding the Substitution Method

The substitution method involves the following key steps:

  1. Solve one equation for one variable: Choose one of the equations and isolate one of the variables (either x or y) on one side of the equation. This will express that variable in terms of the other.
  2. Substitute: Substitute the expression obtained in step 1 into the other equation. This will result in a single equation with only one variable.
  3. Solve the resulting equation: Solve the equation obtained in step 2 for the remaining variable.
  4. Back-substitute: Substitute the value found in step 3 back into either of the original equations (or the expression from step 1) to find the value of the other variable.
  5. Check the solution: Verify that the values obtained for x and y satisfy both original equations.

Evaluating the Options for the Best First Step

Now, let's analyze each of the provided options in the context of our system of equations:

A. Solve for x in the first equation.

If we choose to solve the first equation (5x - 2y = 15) for x, we would need to perform the following steps:

  1. Add 2y to both sides: 5x = 15 + 2y
  2. Divide both sides by 5: x = (15 + 2y) / 5

This results in an expression for x that involves a fraction. While not inherently problematic, dealing with fractions can sometimes increase the complexity of subsequent calculations and introduce opportunities for errors.

B. Solve for y in the first equation.

Solving the first equation for y would involve these steps:

  1. Subtract 5x from both sides: -2y = 15 - 5x
  2. Divide both sides by -2: y = (15 - 5x) / -2

Similar to option A, this also leads to an expression for y that contains a fraction. Therefore, this option might not be the most straightforward starting point.

C. Solve for x in the second equation.

Let's consider solving the second equation (2x + 4y = 12) for x:

  1. Subtract 4y from both sides: 2x = 12 - 4y
  2. Divide both sides by 2: x = (12 - 4y) / 2

Notice that the numerator (12 - 4y) is divisible by 2. We can simplify this expression:

x = 6 - 2y

This gives us a cleaner expression for x without any fractions. This simplification can make the substitution process smoother and less prone to errors.

D. Solve for y in the second equation.

Finally, let's examine solving the second equation for y:

  1. Subtract 2x from both sides: 4y = 12 - 2x
  2. Divide both sides by 4: y = (12 - 2x) / 4

Again, we can simplify this expression since the numerator terms are divisible by 2:

y = (6 - x) / 2

This expression for y still involves a fraction, although a slightly simpler one than in options A and B.

The Optimal First Step

Comparing the results of each option, we can see that option C, solving for x in the second equation, leads to the simplest expression without fractions: x = 6 - 2y. This simplification will make the subsequent substitution and solving steps easier.

Therefore, the best first step in solving the given system of equations using substitution is to solve for x in the second equation.

Why is Solving for x in the Second Equation the Best First Step?

The key reason why solving for x in the second equation is the most strategic first step lies in the resulting expression: x = 6 - 2y. This expression is free of fractions, making it easier to substitute into the first equation and solve for y. Let's illustrate this:

  1. Substitute x = 6 - 2y into the first equation (5x - 2y = 15): 5(6 - 2y) - 2y = 15
  2. Simplify and solve for y: 30 - 10y - 2y = 15 30 - 12y = 15 -12y = -15 y = 5/4
  3. Substitute y = 5/4 back into x = 6 - 2y: x = 6 - 2(5/4) x = 6 - 5/2 x = 7/2

As you can see, starting with the simplified expression x = 6 - 2y made the calculations relatively straightforward. If we had chosen an option that resulted in fractions, the substitution and simplification steps would have been more cumbersome.

General Tips for Choosing the Best First Step in Substitution

Here are some general tips to keep in mind when deciding on the best first step in solving a system of equations using substitution:

  • Look for coefficients of 1 or -1: If one of the variables has a coefficient of 1 or -1 in either equation, it's often easiest to solve for that variable. This avoids introducing fractions in the initial step.
  • Identify easily divisible terms: If the coefficients in one equation have a common factor, solving for a variable in that equation might lead to simplification and avoid fractions.
  • Consider the overall complexity: Evaluate the expressions that would result from solving for each variable in each equation. Choose the option that leads to the simplest expression, even if it doesn't have a coefficient of 1 or -1.
  • Don't be afraid to rearrange: You can rearrange the equations in the system to make the substitution process easier. For example, you might swap the positions of the equations if it makes it clearer which variable to solve for first.

Conclusion

In the given system of equations, the most strategic first step is to solve for x in the second equation. This choice leads to a simplified expression without fractions, making the subsequent steps in the substitution method more manageable. By carefully evaluating the options and considering the resulting expressions, you can optimize your approach to solving systems of equations using substitution.

Remember, practice is key to mastering the substitution method. Work through various examples, and you'll become more adept at identifying the most efficient first steps and navigating the process with confidence.

By understanding the underlying principles and applying these tips, you can effectively use the substitution method to solve a wide range of systems of equations. Always remember to double-check your solutions to ensure accuracy!

This comprehensive guide has hopefully provided you with a clear understanding of how to choose the best first step when using the substitution method to solve systems of equations. Happy problem-solving!