Ernesto's Substitution Errors: A Step-by-Step Guide
Hey guys, let's dive into Ernesto's attempt to solve a system of equations using the substitution method. We'll break down his work step-by-step and pinpoint any errors he might have made along the way. Understanding how to solve systems of equations is super important in math, and the substitution method is a handy tool. So, grab your pencils and let's get started! We'll go through each of Ernesto's steps, explain what he should have done, and highlight where he might have gone wrong. This isn't about criticizing Ernesto, it's about learning from his approach and solidifying our understanding of the substitution method. Let's make sure we've got a solid grasp of how to solve these equations correctly. Systems of equations are collections of two or more equations that we try to solve simultaneously. The solution to a system of equations is the set of values for the variables that satisfy all the equations in the system. The substitution method is one way to find this solution, and it works by solving one equation for one variable and then substituting that expression into the other equation. Are you ready? Let's go!
Understanding the Problem: The System of Equations
Alright, first things first, let's look at the system of equations Ernesto was trying to solve. The system is:
- x - y = 7
- 3x - 2y = 8
This system has two equations and two unknowns: x and y. Our goal is to find the values of x and y that make both equations true at the same time. The substitution method is a clever way to do this. The main idea is to isolate one variable in one equation and then use that information to replace the same variable in the other equation. It is a fantastic method for solving the system of equations.
Now, let's break down how Ernesto approached the problem. By looking at his steps we will determine where the errors were, and we'll learn some best practices to avoid these mistakes. Let's get started. We need to go through each of his steps.
Step-by-Step Analysis of Ernesto's Work
Step 1: Isolating x
Ernesto's first step was:
- x = y + 7
This step is correct! He took the first equation (x - y = 7) and correctly isolated x by adding y to both sides. This is a crucial first step in the substitution method. By expressing x in terms of y, we can substitute this expression into the second equation and solve for y. This is the initial setup; it is a good start. The ability to correctly manipulate equations is the foundation of solving them. Well done, Ernesto, on this one!
Step 2: Substitution
Here's where the substitution takes place:
- 3(y + 7) - 2y = 8
This is also correct. Ernesto took the expression he found for x (which is y + 7) and substituted it into the second equation (3x - 2y = 8). He replaced the x with (y + 7), which gives us the new equation: 3*(y + 7) - 2y = 8. This is the heart of the substitution method. Now, let's see how he simplifies this equation.
Step 3: Distribution
In this step, Ernesto tries to simplify by distributing:
- 3y + 7 - 2y = 8
Oops! This is where the first error appears. When you distribute the 3 across (y + 7), you must multiply both y and 7 by 3. This means that 3 * (y + 7) should become 3y + 21, not 3y + 7. So, the equation should have been: 3y + 21 - 2y = 8. Remember that when distributing, it's essential to multiply every term inside the parentheses by the factor outside. Careful distribution is critical! This one little slip-up can lead to a completely incorrect solution. Always double-check your distribution to avoid these common mistakes. Let's see how this affects the rest of his work.
Step 4: Simplifying and Solving for y
Based on his incorrect distribution, Ernesto did the following:
- y + 7 = 8
Because of the previous error, he correctly simplified. If the equation had been 3y + 21 - 2y = 8, then it would be y + 21 = 8. In this step, he combined like terms, which is correct (3y - 2y = y). Also, he subtracted 7 from both sides to get y = 1. However, since the equation was wrong in the first place, his value for y is wrong. If the previous step had been correct, Ernesto would have correctly combined 3y - 2y and would have then gotten y + 21 = 8. He would have then subtracted 21 from both sides and y = -13.
Step 5: Finding x (The Missing Step)
Ernesto should have then substituted the value of y back into one of the original equations (or into the equation x = y + 7) to find the value of x. Let's see how this would have worked using the correct value of y. If y = -13, then x - (-13) = 7, or x + 13 = 7. Thus, x = -6. The solution to the system of equations is x = -6 and y = -13.
Correcting Ernesto's Work and Finding the Solution
To find the correct solution, let's go through the steps again, correcting the error. First, we have x - y = 7 and 3x - 2y = 8.
- Step 1: x = y + 7 (Correct)
- Step 2: 3(y + 7) - 2y = 8 (Correct Substitution)
- Step 3: 3y + 21 - 2y = 8 (Correct Distribution)
- Step 4: y + 21 = 8 => y = -13 (Solving for y)
- Step 5: x = -13 + 7 = -6 (Solving for x)
The correct solution is x = -6 and y = -13. We can check our answers by plugging them back into the original equations. This is super important for making sure your answer is correct. Let's do that now.
- x - y = 7 => -6 - (-13) = 7 => -6 + 13 = 7 => 7 = 7 (Correct!)
- 3x - 2y = 8 => 3(-6) - 2(-13) = 8 => -18 + 26 = 8 => 8 = 8 (Correct!)
Therefore, our solution x = -6 and y = -13 is correct, and we can be sure of it.
Conclusion: Learning from Mistakes
So, guys, Ernesto made a mistake in the distribution step, but that's okay! We all make mistakes. The important thing is that we learn from them. By carefully examining each step and understanding where the error occurred, we can reinforce our understanding of the substitution method. Remember to double-check your work, especially when distributing, and always check your solutions. Also, make sure that you distribute across all terms inside the parentheses. Keep practicing, and you'll become a pro at solving systems of equations in no time! Keep practicing, and you'll be solving equations like a boss! We are all learning together, and the most important thing is that you learned something new! Way to go!