Solving Systems: Tyler Vs. Han's Substitution Methods

Hey guys! Today, we're diving into a classic math problem: solving a system of equations using the substitution method. We'll break down how Tyler and Han approach the same problem with slightly different initial steps. Buckle up, and let's get started!

The Problem

Tyler and Han are tackling this system of equations:

$$\left\{\begin{array}{l} x + 3y = -5 \\ 9x + 3y = 3 \end{array}\right.$$

Their mission? To find the values of x and y that satisfy both equations simultaneously. There are several ways to solve such systems, but Tyler and Han chose the substitution method. Let's see how their initial moves differ and why both can lead to the correct solution.

Tyler's Approach: Isolating x

Tyler decides to isolate x in the first equation. Here's how he does it:

1. Start with the first equation:
   x + 3y = -5
2. Subtract 3y from both sides to isolate x:
   x = -5 - 3y

So, Tyler gets x = -5 - 3y. Now he's ready to substitute this expression for x into the second equation. Let's walk through the rest of Tyler's steps to show how he would complete the problem.

Completing Tyler's Solution

Now that Tyler has x = -5 - 3y, he substitutes this into the second equation:

1. Second equation:
   9x + 3y = 3
2. Substitute x:
   9(-5 - 3y) + 3y = 3
3. Distribute the 9:
   -45 - 27y + 3y = 3
4. Combine like terms:
   -45 - 24y = 3
5. Add 45 to both sides:
   -24y = 48
6. Divide by -24:
   y = -2

Now that Tyler has the value of y, he can plug it back into the equation x = -5 - 3y to find x:

x = -5 - 3(-2) x = -5 + 6 x = 1

So, Tyler finds that x = 1 and y = -2. This is the solution to the system of equations.

Han's Approach: Isolating 3y

Han, on the other hand, chooses to isolate 3y in the first equation. Here's his approach:

1. Start with the first equation:
   x + 3y = -5
2. Subtract x from both sides to isolate 3y:
   3y = -5 - x

So, Han gets 3y = -5 - x. Now he's ready to substitute this expression for 3y into the second equation. Let’s carry on with Han's solution to see how it pans out.

Completing Han's Solution

Han has 3y = -5 - x. He substitutes this into the second equation:

1. Second equation:
   9x + 3y = 3
2. Substitute 3y:
   9x + (-5 - x) = 3
3. Simplify:
   9x - 5 - x = 3
4. Combine like terms:
   8x - 5 = 3
5. Add 5 to both sides:
   8x = 8
6. Divide by 8:
   x = 1

Now that Han has the value of x, he can plug it back into the equation 3y = -5 - x to find y:

3y = -5 - 1 3y = -6 y = -2

So, Han also finds that x = 1 and y = -2. Both methods arrive at the same solution, which validates their approaches.

Comparing the Two Approaches

• Efficiency: Both approaches are equally valid, but sometimes one might be slightly easier depending on the specific equations. In this case, neither approach has a significant advantage in terms of the number of steps.
• Flexibility: The beauty of the substitution method is its flexibility. You can choose to isolate whichever variable seems easiest to work with. This can depend on the coefficients and constants in the equations.
• Understanding: Both Tyler and Han demonstrate a solid understanding of algebraic manipulation. They correctly apply the rules of algebra to isolate variables and substitute expressions.

Why Substitution Works

The substitution method works because it leverages the idea that if two expressions are equal, one can replace the other without changing the truth of the equation. By expressing one variable in terms of the other, we reduce the system of two equations to a single equation with one variable, which we can then solve. Once we find the value of one variable, we can substitute it back into one of the original equations to find the value of the other variable.

Common Mistakes to Avoid

• Incorrect Substitution: Make sure to substitute the expression into the correct equation and for the correct variable.
• Sign Errors: Pay close attention to signs, especially when distributing or combining like terms.
• Algebraic Errors: Double-check your algebraic manipulations to avoid mistakes in isolating variables or simplifying expressions.

Practice Makes Perfect

Solving systems of equations is a fundamental skill in algebra. The more you practice, the more comfortable you'll become with different methods and strategies. Try solving the following system using both Tyler's and Han's approaches to reinforce your understanding:

$$\left\{\begin{array}{l} 2x + y = 7 \\ x - y = 2 \end{array}\right.$$

Conclusion

Whether you choose to isolate x, y, or even a multiple of a variable like 3y, the substitution method is a powerful tool for solving systems of equations. Tyler's and Han's approaches show that there's often more than one way to tackle a problem, and understanding the underlying principles allows you to choose the most efficient path. Keep practicing, and you'll become a system-solving pro in no time! Keep up the great work, guys!