Use The Substitution Method To Solve The System Of Equations.${ \begin{align*} 5x + 2y &= 1 \ y &= -x + 2 \end{align*} }$A. { (-3, 8)$}$ B. { (-1, 3)$}$ C. { (-2, 4)$}$ D. { (-3, 5)$}$
Introduction
Solving systems of equations is a fundamental concept in mathematics, and it is essential to understand various methods to solve them. One of the most popular methods is the substitution method, which involves substituting one equation into another to solve for the variables. In this article, we will use the substitution method to solve a system of equations and provide a step-by-step guide on how to do it.
What is the Substitution Method?
The substitution method is a technique used to solve systems of equations by substituting one equation into another. This method is useful when one of the equations is already solved for one of the variables. The substitution method involves the following steps:
- Identify one of the equations that is already solved for one of the variables.
- Substitute the expression for the variable into the other equation.
- Solve the resulting equation for the other variable.
- Substitute the value of the variable back into one of the original equations to find the value of the other variable.
Step-by-Step Guide to Solving the System of Equations
Now, let's use the substitution method to solve the system of equations:
{ \begin{align*} 5x + 2y &= 1 \\ y &= -x + 2 \end{align*} \}
Step 1: Identify the Equation Already Solved for One of the Variables
In this system of equations, the second equation is already solved for the variable y:
y = -x + 2
Step 2: Substitute the Expression for the Variable into the Other Equation
Substitute the expression for y into the first equation:
5x + 2(-x + 2) = 1
Step 3: Simplify the Equation
Expand and simplify the equation:
5x - 2x + 4 = 1
Combine like terms:
3x + 4 = 1
Step 4: Solve for the Variable
Subtract 4 from both sides of the equation:
3x = -3
Divide both sides of the equation by 3:
x = -1
Step 5: Substitute the Value of the Variable Back into One of the Original Equations
Substitute the value of x back into the second equation:
y = -(-1) + 2
Simplify the equation:
y = 1 + 2
y = 3
Conclusion
Using the substitution method, we have solved the system of equations and found the values of x and y:
x = -1
y = 3
Therefore, the solution to the system of equations is:
(-1, 3)
Answer
The correct answer is:
B. (-1, 3)
Discussion
Q: What is the substitution method?
A: The substitution method is a technique used to solve systems of equations by substituting one equation into another. This method is useful when one of the equations is already solved for one of the variables.
Q: When should I use the substitution method?
A: You should use the substitution method when one of the equations is already solved for one of the variables. This method is particularly useful when you have a linear equation that is already solved for one of the variables.
Q: How do I know which equation to substitute into the other equation?
A: You should substitute the equation that is already solved for one of the variables into the other equation. This will allow you to solve for the other variable.
Q: What if I have two equations that are not linear? Can I still use the substitution method?
A: No, you cannot use the substitution method if you have two equations that are not linear. The substitution method only works for linear equations.
Q: Can I use the substitution method to solve a system of equations with three or more variables?
A: No, the substitution method is only used to solve systems of equations with two variables. If you have a system of equations with three or more variables, you will need to use a different method, such as the elimination method or the graphing method.
Q: How do I know if I have made a mistake when using the substitution method?
A: If you have made a mistake when using the substitution method, you may end up with an equation that is not true. For example, if you substitute an equation into another equation and get a result that is not true, you may have made a mistake.
Q: Can I use the substitution method to solve a system of equations with fractions or decimals?
A: Yes, you can use the substitution method to solve a system of equations with fractions or decimals. However, you will need to follow the same steps as you would with linear equations.
Q: How do I check my answer when using the substitution method?
A: To check your answer, you should substitute the values of x and y back into both original equations to make sure they are true.
Q: What if I get a different answer when using the substitution method than I do when using another method?
A: If you get a different answer when using the substitution method than you do when using another method, you may have made a mistake. You should recheck your work and make sure you have followed the correct steps.
Conclusion
The substitution method is a powerful tool for solving systems of equations. By following the steps outlined in this article, you can use the substitution method to solve systems of equations and find the values of x and y. Remember to check your answer by substituting the values of x and y back into both original equations to make sure they are true.
Additional Resources
If you are having trouble with the substitution method, you may want to try the following resources:
- Online tutorials and videos
- Math textbooks and workbooks
- Online math communities and forums
- Math tutors or teachers
Final Thoughts
Solving systems of equations is an important skill to have in mathematics. By using the substitution method, you can solve systems of equations and find the values of x and y. Remember to follow the steps outlined in this article and to check your answer by substituting the values of x and y back into both original equations to make sure they are true.