Select The Correct Answer.Solve The System Of Equations Below.${ \begin{array}{l} 5x + 2y = 9 \ 2x - 3y = 15 \end{array} }$A. { (-3, 3)$}$ B. { (12, -3)$}$ C. { (3, -3)$}$ D. { (-3, 12)$}$
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Introduction
Solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the solution to a set of two or more linear equations, where each equation is in the form of ax + by = c. In this article, we will focus on solving a system of two linear equations using the method of substitution and elimination.
The System of Equations
The given system of equations is:
{ \begin{array}{l} 5x + 2y = 9 \\ 2x - 3y = 15 \end{array} \}
Method of Substitution
One way to solve this system of equations is by using the method of substitution. This involves solving one equation for one variable and then substituting that expression into the other equation.
Step 1: Solve the First Equation for x
Let's solve the first equation for x:
5x + 2y = 9
Subtract 2y from both sides:
5x = 9 - 2y
Divide both sides by 5:
x = (9 - 2y) / 5
Step 2: Substitute the Expression for x into the Second Equation
Now, substitute the expression for x into the second equation:
2((9 - 2y) / 5) - 3y = 15
Multiply both sides by 5 to eliminate the fraction:
2(9 - 2y) - 15y = 75
Expand and simplify:
18 - 4y - 15y = 75
Combine like terms:
-19y = 57
Divide both sides by -19:
y = -57 / 19
y = -3
Step 3: Find the Value of x
Now that we have the value of y, substitute it back into the expression for x:
x = (9 - 2(-3)) / 5
Simplify:
x = (9 + 6) / 5
x = 15 / 5
x = 3
Method of Elimination
Another way to solve this system of equations is by using the method of elimination. This involves adding or subtracting the equations to eliminate one variable.
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one variable, we need to multiply the equations by necessary multiples such that the coefficients of y's in both equations are the same:
5x + 2y = 9
Multiply the first equation by 3:
15x + 6y = 27
2x - 3y = 15
Multiply the second equation by 2:
4x - 6y = 30
Step 2: Add the Equations
Now, add the two equations:
(15x + 6y) + (4x - 6y) = 27 + 30
Combine like terms:
19x = 57
Divide both sides by 19:
x = 57 / 19
x = 3
Step 3: Find the Value of y
Now that we have the value of x, substitute it back into one of the original equations to find the value of y:
5x + 2y = 9
Substitute x = 3:
5(3) + 2y = 9
Simplify:
15 + 2y = 9
Subtract 15 from both sides:
2y = -6
Divide both sides by 2:
y = -6 / 2
y = -3
Conclusion
In this article, we solved a system of two linear equations using the method of substitution and elimination. We found that the solution to the system is x = 3 and y = -3.
Answer
The correct answer is:
C. (3, -3)
This solution satisfies both equations:
5(3) + 2(-3) = 15 - 6 = 9
2(3) - 3(-3) = 6 + 9 = 15
Therefore, the correct answer is (3, -3).
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Introduction
Solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. In our previous article, we discussed the method of substitution and elimination to solve a system of two linear equations. In this article, we will provide a Q&A guide to help you better understand the concept and solve systems of linear equations.
Q: What is a system of linear equations?
A system of linear equations is a set of two or more linear equations, where each equation is in the form of ax + by = c.
A: How do I know if a system of linear equations has a solution?
A system of linear equations has a solution if the two equations are consistent, meaning they have the same solution. If the equations are inconsistent, they have no solution.
Q: What is the difference between the method of substitution and elimination?
The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one variable.
A: How do I choose between the method of substitution and elimination?
You can choose between the two methods based on the coefficients of the variables in the equations. If the coefficients of one variable are the same, you can use the method of elimination. If the coefficients of one variable are different, you can use the method of substitution.
Q: What if I have a system of three or more linear equations?
If you have a system of three or more linear equations, you can use the method of substitution or elimination to solve the system. However, you may need to use a more advanced method, such as Gaussian elimination or matrix operations.
A: How do I check if my solution is correct?
To check if your solution is correct, you can substitute the values of the variables back into the original equations and verify that they are true.
Q: What if I have a system of linear equations with no solution?
If you have a system of linear equations with no solution, it means that the equations are inconsistent. This can happen if the equations are contradictory, meaning they cannot be true at the same time.
A: How do I know if a system of linear equations has infinitely many solutions?
A system of linear equations has infinitely many solutions if the equations are dependent, meaning they are multiples of each other.
Q: Can I use a calculator to solve systems of linear equations?
Yes, you can use a calculator to solve systems of linear equations. Many calculators have built-in functions to solve systems of linear equations, such as the "solve" function.
A: How do I graph a system of linear equations?
To graph a system of linear equations, you can use a graphing calculator or a graphing software. You can also use a coordinate plane to plot the equations and find the point of intersection.
Conclusion
Solving systems of linear equations is a fundamental concept in mathematics, and it requires a good understanding of algebra and geometry. By following the steps outlined in this Q&A guide, you can solve systems of linear equations and understand the concept better.
Frequently Asked Questions
- Q: What is a system of linear equations?
- A: A system of linear equations is a set of two or more linear equations, where each equation is in the form of ax + by = c.
- Q: How do I know if a system of linear equations has a solution?
- A: A system of linear equations has a solution if the two equations are consistent, meaning they have the same solution.
- Q: What is the difference between the method of substitution and elimination?
- A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one variable.
Additional Resources
- Khan Academy: Systems of Linear Equations
- Mathway: Systems of Linear Equations
- Wolfram Alpha: Systems of Linear Equations
Answer Key
- Q: What is a system of linear equations?
- A: A system of linear equations is a set of two or more linear equations, where each equation is in the form of ax + by = c.
- Q: How do I know if a system of linear equations has a solution?
- A: A system of linear equations has a solution if the two equations are consistent, meaning they have the same solution.
- Q: What is the difference between the method of substitution and elimination?
- A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one variable.