Which Ordered Pair Below Is A Solution To The Following System Of Equations?1. $5x - Y = 2$2. X − 0.5 Y = 3 X - 0.5y = 3 X − 0.5 Y = 3 A. (-4, 4)
Introduction
Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore how to solve a system of linear equations using the substitution method and the elimination method. We will also examine a specific system of equations and determine which ordered pair is a solution to the system.
What are Systems of Linear Equations?
A system of linear equations is a set of two or more linear equations that are solved simultaneously. Each equation in the system is a linear equation, which means it can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are variables.
The Substitution Method
The substitution method is a technique used to solve systems of linear equations. This method involves solving one equation for one variable and then substituting that expression into the other equation.
Step 1: Solve One Equation for One Variable
Let's consider the first equation:
5x - y = 2
We can solve this equation for y:
y = 5x - 2
Step 2: Substitute the Expression into the Other Equation
Now, let's substitute the expression for y into the second equation:
x - 0.5(5x - 2) = 3
Step 3: Simplify the Equation
Simplifying the equation, we get:
x - 2.5x + 1 = 3
Combine like terms:
-1.5x + 1 = 3
Subtract 1 from both sides:
-1.5x = 2
Divide both sides by -1.5:
x = -4/3
Step 4: Find the Value of y
Now that we have the value of x, we can find the value of y by substituting x into one of the original equations. Let's use the first equation:
5x - y = 2
Substitute x = -4/3:
5(-4/3) - y = 2
Simplify the equation:
-20/3 - y = 2
Multiply both sides by 3:
-20 - 3y = 6
Add 20 to both sides:
-3y = 26
Divide both sides by -3:
y = -26/3
The Elimination Method
The elimination method is another technique used to solve systems of linear equations. This method 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. Let's multiply the first equation by 0.5 and the second equation by 1:
0.5(5x - y) = 0.5(2)
x - 0.5y = 3
Step 2: Add or Subtract the Equations
Now, let's add the two equations:
(5x - y) + (x - 0.5y) = 2 + 3
Combine like terms:
6x - 1.5y = 5
Step 3: Solve for One Variable
Now that we have the new equation, we can solve for one variable. Let's solve for x:
6x = 5 + 1.5y
Divide both sides by 6:
x = (5 + 1.5y)/6
Step 4: Find the Value of y
Now that we have the value of x, we can find the value of y by substituting x into one of the original equations. Let's use the first equation:
5x - y = 2
Substitute x = (5 + 1.5y)/6:
5((5 + 1.5y)/6) - y = 2
Simplify the equation:
(25 + 7.5y)/6 - y = 2
Multiply both sides by 6:
25 + 7.5y - 6y = 12
Combine like terms:
1.5y = -13
Divide both sides by 1.5:
y = -13/1.5
y = -26/3
Which Ordered Pair is a Solution to the System?
Now that we have solved the system of equations using both the substitution method and the elimination method, we can determine which ordered pair is a solution to the system.
The ordered pair (-4, 4) is a solution to the system if it satisfies both equations.
Let's check if the ordered pair (-4, 4) satisfies the first equation:
5x - y = 2
Substitute x = -4 and y = 4:
5(-4) - 4 = 2
-20 - 4 = 2
-24 ≠ 2
The ordered pair (-4, 4) does not satisfy the first equation.
However, let's check if the ordered pair (-4, 4) satisfies the second equation:
x - 0.5y = 3
Substitute x = -4 and y = 4:
-4 - 0.5(4) = 3
-4 - 2 = 3
-6 ≠ 3
The ordered pair (-4, 4) does not satisfy the second equation.
Therefore, the ordered pair (-4, 4) is not a solution to the system of equations.
Conclusion
In this article, we have explored how to solve a system of linear equations using the substitution method and the elimination method. We have also examined a specific system of equations and determined which ordered pair is a solution to the system. The ordered pair (-4, 4) is not a solution to the system of equations.
References
- [1] "Systems of Linear Equations" by Math Open Reference
- [2] "Solving Systems of Linear Equations" by Khan Academy
- [3] "The Substitution Method" by Purplemath
- [4] "The Elimination Method" by Mathway
Additional Resources
- [1] "Systems of Linear Equations" by MIT OpenCourseWare
- [2] "Solving Systems of Linear Equations" by Wolfram Alpha
- [3] "The Substitution Method" by IXL
- [4] "The Elimination Method" by Symbolab
Frequently Asked Questions (FAQs) about Systems of Linear Equations ====================================================================
Q: What is a system of linear equations?
A: A system of linear equations is a set of two or more linear equations that are solved simultaneously. Each equation in the system is a linear equation, which means it can be written in the form:
ax + by = c
where a, b, and c are constants, and x and y are variables.
Q: How do I solve a system of linear equations?
A: There are two main methods to solve a system of linear equations: the substitution method and the elimination method.
- The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
- The elimination method involves adding or subtracting the equations to eliminate one variable.
Q: What is the substitution method?
A: The substitution method is a technique used to solve systems of linear equations. This method involves solving one equation for one variable and then substituting that expression into the other equation.
Q: What is the elimination method?
A: The elimination method is another technique used to solve systems of linear equations. This method involves adding or subtracting the equations to eliminate one variable.
Q: How do I determine which method to use?
A: To determine which method to use, you can try the following:
- If one equation is already solved for one variable, use the substitution method.
- If the coefficients of one variable are the same in both equations, use the elimination method.
Q: What is the difference between a solution and a solution set?
A: A solution is a specific value of x and y that satisfies both equations in a system of linear equations. A solution set is the set of all possible solutions to a system of linear equations.
Q: How do I find the solution set of a system of linear equations?
A: To find the solution set of a system of linear equations, you can use the following steps:
- Solve the system of linear equations using either the substitution method or the elimination method.
- Write the solution in the form (x, y).
- If the solution is a single point, the solution set is a single point.
- If the solution is a line, the solution set is a line.
- If the solution is a plane, the solution set is a plane.
Q: What is the importance of solving systems of linear equations?
A: Solving systems of linear equations is an important skill in mathematics and has many real-world applications, such as:
- Modeling real-world problems
- Solving optimization problems
- Finding the intersection of two or more lines or planes
Q: How do I practice solving systems of linear equations?
A: To practice solving systems of linear equations, you can try the following:
- Use online resources, such as Khan Academy or Mathway, to practice solving systems of linear equations.
- Work on problems from a textbook or online resource.
- Try to solve systems of linear equations on your own, without looking at the solution.
Q: What are some common mistakes to avoid when solving systems of linear equations?
A: Some common mistakes to avoid when solving systems of linear equations include:
- Not following the order of operations
- Not checking the solution to make sure it satisfies both equations
- Not using the correct method to solve the system of linear equations
Q: How do I check my solution to a system of linear equations?
A: To check your solution to a system of linear equations, you can try the following:
- Substitute the solution into both equations and check if it satisfies both equations.
- Use a graphing calculator or online resource to check if the solution is correct.
Q: What are some real-world applications of solving systems of linear equations?
A: Some real-world applications of solving systems of linear equations include:
- Modeling population growth
- Solving optimization problems
- Finding the intersection of two or more lines or planes
Q: How do I use technology to solve systems of linear equations?
A: To use technology to solve systems of linear equations, you can try the following:
- Use a graphing calculator to graph the equations and find the intersection.
- Use online resources, such as Khan Academy or Mathway, to solve systems of linear equations.
- Use a computer algebra system, such as Mathematica or Maple, to solve systems of linear equations.