Ernesto Tried To Determine The Solution For The System Of Equations Using Substitution. His Work Is Shown Below.$ \begin{array}{r} x-y=7 \ 3x-2y=8 \end{array} }$Step 1 { X = Y + 7$ $Step 2: ${ 3(y + 7) - 2y = 8\$} Step
Introduction
In mathematics, solving systems of equations is a fundamental concept that involves finding the values of variables that satisfy multiple equations simultaneously. One of the methods used to solve systems of equations is substitution, where one equation is solved for one variable, and then the value of that variable is substituted into the other equation. In this article, we will analyze Ernesto's attempt at solving a system of equations using substitution and discuss the correct steps to follow.
Ernesto's Work
Ernesto tried to determine the solution for the system of equations using substitution. His work is shown below:
Step 1: Solving the First Equation for x
Ernesto started by solving the first equation for x:
This is a correct step, as it isolates the variable x in terms of y.
Step 2: Substituting x into the Second Equation
Ernesto then substituted x into the second equation:
This is also a correct step, as it substitutes the expression for x from the first equation into the second equation.
Discussion
However, Ernesto's work is incomplete, and he did not solve for y. To find the value of y, we need to simplify the equation obtained in Step 2.
Simplifying the Equation
Let's simplify the equation obtained in Step 2:
Expanding the left-hand side of the equation, we get:
Combining like terms, we get:
Subtracting 21 from both sides of the equation, we get:
Finding the Value of x
Now that we have the value of y, we can substitute it into the expression for x obtained in Step 1:
Substituting y = -13, we get:
Simplifying the expression, we get:
Conclusion
In conclusion, Ernesto's attempt at solving a system of equations using substitution was correct up to a point, but he did not complete the solution. By simplifying the equation obtained in Step 2, we were able to find the value of y, and then substitute it into the expression for x to find the value of x. This demonstrates the importance of following the correct steps when solving systems of equations using substitution.
Tips for Solving Systems of Equations using Substitution
Here are some tips for solving systems of equations using substitution:
- Make sure to isolate one variable in one of the equations.
- Substitute the expression for the isolated variable into the other equation.
- Simplify the resulting equation to find the value of the other variable.
- Substitute the value of the other variable back into one of the original equations to find the value of the isolated variable.
By following these tips, you can successfully solve systems of equations using substitution.
Common Mistakes to Avoid
Here are some common mistakes to avoid when solving systems of equations using substitution:
- Failing to isolate one variable in one of the equations.
- Not simplifying the resulting equation to find the value of the other variable.
- Not substituting the value of the other variable back into one of the original equations to find the value of the isolated variable.
By avoiding these mistakes, you can ensure that your solution is correct and complete.
Real-World Applications
Solving systems of equations using substitution has many real-world applications, including:
- Physics: Solving systems of equations is used to describe the motion of objects in physics.
- Engineering: Solving systems of equations is used to design and optimize systems in engineering.
- Economics: Solving systems of equations is used to model economic systems and make predictions about economic outcomes.
By understanding how to solve systems of equations using substitution, you can apply this knowledge to real-world problems and make informed decisions.
Conclusion
Q: What is substitution in solving systems of equations?
A: Substitution is a method used to solve systems of equations by substituting the expression for one variable from one equation into the other equation.
Q: How do I know which variable to isolate first?
A: You can choose any variable to isolate first, but it's often easier to isolate the variable that appears in both equations.
Q: What if I get a quadratic equation after substituting?
A: If you get a quadratic equation after substituting, you can try to factor it or use the quadratic formula to solve for the variable.
Q: Can I use substitution to solve systems of equations with more than two variables?
A: Yes, you can use substitution to solve systems of equations with more than two variables. However, it may be more complicated and require more steps.
Q: What if I make a mistake in my substitution?
A: If you make a mistake in your substitution, you may end up with an incorrect equation. Double-check your work and make sure you're substituting correctly.
Q: Can I use substitution to solve systems of equations with fractions or decimals?
A: Yes, you can use substitution to solve systems of equations with fractions or decimals. Just make sure to simplify your equations and follow the correct steps.
Q: How do I know if my solution is correct?
A: To check if your solution is correct, substitute the values of the variables back into the original equations and make sure they're true.
Q: What if I get a system of equations with no solution?
A: If you get a system of equations with no solution, it means that the equations are inconsistent and there's no value of the variables that can satisfy both equations.
Q: Can I use substitution to solve systems of equations with absolute values or inequalities?
A: Yes, you can use substitution to solve systems of equations with absolute values or inequalities. However, it may require more steps and careful handling of the absolute value or inequality.
Q: What are some common mistakes to avoid when using substitution?
A: Some common mistakes to avoid when using substitution include:
- Failing to isolate one variable in one of the equations.
- Not simplifying the resulting equation to find the value of the other variable.
- Not substituting the value of the other variable back into one of the original equations to find the value of the isolated variable.
- Making a mistake in the substitution process.
Q: How can I practice using substitution to solve systems of equations?
A: You can practice using substitution to solve systems of equations by working through examples and exercises in your textbook or online resources. You can also try creating your own systems of equations and solving them using substitution.
Q: What are some real-world applications of solving systems of equations using substitution?
A: Solving systems of equations using substitution has many real-world applications, including:
- Physics: Solving systems of equations is used to describe the motion of objects in physics.
- Engineering: Solving systems of equations is used to design and optimize systems in engineering.
- Economics: Solving systems of equations is used to model economic systems and make predictions about economic outcomes.
- Computer Science: Solving systems of equations is used in computer science to solve problems in graph theory and network analysis.
By understanding how to solve systems of equations using substitution, you can apply this knowledge to real-world problems and make informed decisions.