Use The Substitution Method To Solve The System Of Equations. Choose The Correct Ordered Pair.${ \begin{align*} 3x - Y &= 7 \ 2x - 2y &= 2 \end{align*} }$A. { (3, -1)$}$ B. { (2, 2)$}$ C. { (3, 2)$}$ D.
Introduction
Solving systems of equations is a fundamental concept in algebra, and there are several methods to approach it. In this article, we will focus on the substitution method, which involves solving one equation for a variable and then substituting that expression into the other equation. This method is particularly useful when one of the equations is linear and the other is quadratic or has a more complex form.
The Substitution Method
The substitution method involves solving one equation for a variable and then substituting that expression into the other equation. To begin, we need to identify which equation to solve first. In this case, we will solve the first equation for y:
We can isolate y by subtracting 3x from both sides and then multiplying both sides by -1:
Now that we have an expression for y, we can substitute this expression into the second equation:
Substituting y = 3x - 7 into the second equation, we get:
Expanding and simplifying the equation, we get:
Combine like terms:
Subtract 14 from both sides:
Divide both sides by -4:
Now that we have the value of x, we can substitute this value back into one of the original equations to find the value of y. We will use the first equation:
Substituting x = 3, we get:
Expanding and simplifying the equation, we get:
Subtract 9 from both sides:
Multiply both sides by -1:
Therefore, the solution to the system of equations is (3, 2).
Choosing the Correct Ordered Pair
Now that we have solved the system of equations, we need to choose the correct ordered pair from the options provided. Let's review the options:
A. (3, -1) B. (2, 2) C. (3, 2) D. (4, 3)
Based on our solution, we can see that the correct ordered pair is (3, 2), which corresponds to option C.
Conclusion
Q: What is the substitution method?
A: The substitution method is a technique used to solve systems of equations by solving one equation for a variable and then substituting that expression into the other equation.
Q: When should I use the substitution method?
A: You should use the substitution method when one of the equations is linear and the other is quadratic or has a more complex form. This method is particularly useful when you need to isolate a variable in one of the equations.
Q: How do I choose which equation to solve first?
A: To choose which equation to solve first, look for the equation that is easiest to solve. If one equation has a variable with a coefficient of 1, it is often easier to solve that equation first.
Q: What if I get stuck during the substitution process?
A: If you get stuck during the substitution process, try simplifying the equation by combining like terms or using algebraic properties such as the distributive property.
Q: Can I use the substitution method with systems of equations that have more than two equations?
A: Yes, you can use the substitution method with systems of equations that have more than two equations. However, you will need to solve the system of equations in pairs, using the substitution method to solve each pair of equations.
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. This can happen if the equations are contradictory, such as 2x + 3 = 5 and 2x + 3 = 7.
Q: Can I use the substitution method with systems of equations that have variables with fractional coefficients?
A: Yes, you can use the substitution method with systems of equations that have variables with fractional coefficients. However, you will need to be careful when simplifying the equation to avoid introducing extraneous solutions.
Q: What if I get a system of equations with infinitely many solutions?
A: If you get a system of equations with infinitely many solutions, it means that the equations are dependent. This can happen if the equations are equivalent, such as 2x + 3 = 5 and x + 3/2 = 5/2.
Q: Can I use the substitution method with systems of equations that have absolute value expressions?
A: Yes, you can use the substitution method with systems of equations that have absolute value expressions. However, you will need to be careful when simplifying the equation to avoid introducing extraneous solutions.
Conclusion
In this article, we answered some frequently asked questions about solving systems of equations using the substitution method. We covered topics such as choosing which equation to solve first, simplifying equations, and dealing with systems of equations that have no solution, infinitely many solutions, or variables with fractional coefficients.