How to Determine if an Ordered Pair is a Solution to an Equation

Oct 13, 2025 by talynt.org 55 views

Understanding the Equation

The given equation is 6x - \frac{y}{2} = 14. To determine which of the ordered pairs (x, y) is a solution to this equation, we need to substitute each pair into the equation and check if it holds true.

Substituting the First Ordered Pair (-9, 3)

Let's start by substituting the first ordered pair (-9, 3) into the equation.

6x - \frac{y}{2} = 14

Substituting x = -9 and y = 3, we get:

6(-9) - \frac{3}{2} = 14

-54 - \frac{3}{2} = 14

-54 - 1.5 = 14

-55.5 = 14

This is not true, so the first ordered pair (-9, 3) is not a solution to the equation.

Substituting the Second Ordered Pair (2, -4)

Now, let's substitute the second ordered pair (2, -4) into the equation.

6x - \frac{y}{2} = 14

Substituting x = 2 and y = -4, we get:

6(2) - \frac{-4}{2} = 14

12 + 2 = 14

14 = 14

This is true, so the second ordered pair (2, -4) is a solution to the equation.

Conclusion

Based on the substitutions, we can conclude that the second ordered pair (2, -4) is a solution to the equation 6x - \frac{y}{2} = 14, but the first ordered pair (-9, 3) is not.

Why is the First Ordered Pair Not a Solution?

The first ordered pair (-9, 3) is not a solution to the equation because when we substitute x = -9 and y = 3 into the equation, we get -55.5 = 14, which is not true. This means that the equation does not hold for the first ordered pair.

Why is the Second Ordered Pair a Solution?

The second ordered pair (2, -4) is a solution to the equation because when we substitute x = 2 and y = -4 into the equation, we get 14 = 14, which is true. This means that the equation holds for the second ordered pair.

What Does it Mean to be a Solution to an Equation?

To be a solution to an equation, an ordered pair (x, y) must satisfy the equation when substituted into it. In other words, when we plug the values of x and y into the equation, the equation must hold true.

How to Determine if an Ordered Pair is a Solution to an Equation

To determine if an ordered pair is a solution to an equation, we need to substitute the values of x and y into the equation and check if it holds true. If the equation holds true, then the ordered pair is a solution to the equation.

Real-World Applications of Solving Equations

Solving equations has many real-world applications, such as:

Physics and Engineering: Equations are used to describe the motion of objects, the behavior of electrical circuits, and the properties of materials.
Computer Science: Equations are used to model complex systems, optimize algorithms, and solve problems in machine learning and data analysis.
Economics: Equations are used to model economic systems, predict market trends, and optimize resource allocation.

Conclusion

In conclusion, the second ordered pair (2, -4) is a solution to the equation 6x - \frac{y}{2} = 14, but the first ordered pair (-9, 3) is not. To determine if an ordered pair is a solution to an equation, we need to substitute the values of x and y into the equation and check if it holds true. Solving equations has many real-world applications, and it is an essential tool in many fields, including physics, computer science, and economics.

Q: What is an equation?

A: An equation is a statement that two mathematical expressions are equal. It is a way of expressing a relationship between variables and constants.

Q: What is a solution to an equation?

A: A solution to an equation is an ordered pair (x, y) that satisfies the equation when substituted into it. In other words, when we plug the values of x and y into the equation, the equation must hold true.

Q: How do I determine if an ordered pair is a solution to an equation?

A: To determine if an ordered pair is a solution to an equation, we need to substitute the values of x and y into the equation and check if it holds true. If the equation holds true, then the ordered pair is a solution to the equation.

Q: What is the difference between a solution and a value?

A: A solution is an ordered pair (x, y) that satisfies the equation, while a value is a single number that satisfies the equation. For example, if we have the equation x + 2 = 5, the solution is (3, 0), but the value is 3.

Q: Can an equation have multiple solutions?

A: Yes, an equation can have multiple solutions. For example, if we have the equation x^2 = 4, the solutions are (2, 0) and (-2, 0).

Q: Can an equation have no solutions?

A: Yes, an equation can have no solutions. For example, if we have the equation x^2 = -4, there are no real solutions, because the square of any real number is non-negative.

Q: How do I solve a linear equation?

A: To solve a linear equation, we need to isolate the variable on one side of the equation. We can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, we need to use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. This formula gives us two solutions for the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that gives us the solutions to a quadratic equation. It is: x = (-b ± √(b^2 - 4ac)) / 2a.

Q: Can I use the quadratic formula to solve any quadratic equation?

A: Yes, you can use the quadratic formula to solve any quadratic equation. However, if the discriminant (b^2 - 4ac) is negative, the equation has no real solutions.

Q: What is the discriminant?

A: The discriminant is the expression b^2 - 4ac in the quadratic formula. It tells us whether the equation has real solutions or not.

Q: Can I use the quadratic formula to solve equations with complex solutions?

A: Yes, you can use the quadratic formula to solve equations with complex solutions. However, you need to be careful when working with complex numbers.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation of the form ax + b = c, where a, b, and c are constants. A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: Can I use the quadratic formula to solve equations with higher degrees?

A: No, you cannot use the quadratic formula to solve equations with higher degrees. The quadratic formula is only applicable to quadratic equations.

Q: What is the next step after solving an equation?

A: After solving an equation, the next step is to check your solutions by plugging them back into the original equation. This ensures that your solutions are correct.

Q: Can I use a calculator to solve equations?

A: Yes, you can use a calculator to solve equations. However, you need to be careful when using a calculator, as it may give you incorrect solutions.

Q: What is the importance of solving equations?

A: Solving equations is an essential skill in mathematics and has many real-world applications. It helps us model and solve problems in physics, engineering, computer science, and economics.

Q: Can I use solving equations to solve problems in real life?

A: Yes, you can use solving equations to solve problems in real life. For example, you can use equations to model the motion of objects, the behavior of electrical circuits, and the properties of materials.

Q: What are some common mistakes to avoid when solving equations?

A: Some common mistakes to avoid when solving equations include:

Not checking your solutions
Not using the correct formula or method
Not being careful with signs and symbols
Not checking for extraneous solutions

Q: How can I practice solving equations?

A: You can practice solving equations by:

Working on exercises and problems
Using online resources and tools
Joining a study group or finding a study partner
Asking your teacher or tutor for help

Q: What are some resources available to help me solve equations?

A: Some resources available to help you solve equations include:

Online tutorials and videos
Math textbooks and workbooks
Online calculators and software
Math apps and games

Q: Can I use technology to help me solve equations?

A: Yes, you can use technology to help you solve equations. For example, you can use online calculators, math software, and math apps to solve equations and check your solutions.

Q: What are some tips for solving equations?

A: Some tips for solving equations include:

Read the problem carefully and understand what is being asked
Use the correct formula or method
Check your solutions carefully
Be careful with signs and symbols
Use technology to help you solve equations

Q: Can I use solving equations to solve problems in other subjects?

A: Yes, you can use solving equations to solve problems in other subjects, such as physics, engineering, computer science, and economics. Solving equations is an essential skill that has many real-world applications.