15. Which Ordered Pair Is A Solution To The Equation $x + Y = 14$?A. (3, 10) B. (6, 8) C. (4.5, 4.5) D. (2.7, 9.3)

Introduction

In mathematics, linear equations are a fundamental concept that helps us understand the relationship between variables. One of the key aspects of linear equations is finding ordered pairs that satisfy the equation. In this article, we will explore how to solve linear equations and find ordered pairs that satisfy the equation $x + y = 14$.

What are Ordered Pairs?

An ordered pair is a pair of numbers that are written in a specific order, usually in the form $(x, y)$. For example, the ordered pair $(3, 4)$ represents the values of $x$ and $y$ as $3$ and $4$, respectively. Ordered pairs are used to represent points on a coordinate plane and are a crucial concept in mathematics.

Understanding the Equation $x + y = 14$

The equation $x + y = 14$ is a linear equation that represents the relationship between the variables $x$ and $y$. To find the ordered pair that satisfies this equation, we need to find the values of $x$ and $y$ that make the equation true. In other words, we need to find the values of $x$ and $y$ that add up to $14$.

Analyzing the Options

Let's analyze the options given to us:

• A. (3, 10)
• B. (6, 8)
• C. (4.5, 4.5)
• D. (2.7, 9.3)

We need to determine which of these ordered pairs satisfies the equation $x + y = 14$.

Option A: (3, 10)

Let's substitute the values of $x$ and $y$ from option A into the equation:

$3 + 10 = 13$

Since $13$ is not equal to $14$, option A is not a solution to the equation.

Option B: (6, 8)

Let's substitute the values of $x$ and $y$ from option B into the equation:

$6 + 8 = 14$

Since $14$ is equal to $14$, option B is a solution to the equation.

Option C: (4.5, 4.5)

Let's substitute the values of $x$ and $y$ from option C into the equation:

$4.5 + 4.5 = 9$

Since $9$ is not equal to $14$, option C is not a solution to the equation.

Option D: (2.7, 9.3)

Let's substitute the values of $x$ and $y$ from option D into the equation:

$2.7 + 9.3 = 12$

Since $12$ is not equal to $14$, option D is not a solution to the equation.

Conclusion

In conclusion, the ordered pair that satisfies the equation $x + y = 14$ is option B: (6, 8). This means that the values of $x$ and $y$ that add up to $14$ are $6$ and $8$, respectively.

Tips and Tricks

• When solving linear equations, it's essential to substitute the values of $x$ and $y$ into the equation to determine if they satisfy the equation.
• Use the order of operations (PEMDAS) to simplify the equation and make it easier to solve.
• Check your work by plugging the values of $x$ and $y$ back into the equation to ensure that they satisfy the equation.

Practice Problems

Try solving the following linear equations to practice your skills:

• $x + y = 20$
• $x - y = 5$
• $2x + 3y = 12$

Real-World Applications

Linear equations have numerous real-world applications, including:

• Finance: Linear equations are used to calculate interest rates, investment returns, and other financial metrics.
• Science: Linear equations are used to model population growth, chemical reactions, and other scientific phenomena.
• Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Conclusion

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Introduction

In our previous article, we explored how to solve linear equations and find ordered pairs that satisfy the equation $x + y = 14$. In this article, we will answer some frequently asked questions about solving linear equations and provide additional tips and tricks to help you become proficient in this area.

Q: What is a linear equation?

A: A linear equation is an equation that 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 linear equation?

A: To solve a linear equation, you need to isolate the variable (either $x$ or $y$) on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

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

A: A linear equation is an equation that can be written in the form $ax + by = c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables. A quadratic equation, on the other hand, is an equation that can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is a variable.

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to find two points on the line and plot them on a coordinate plane. You can then draw a line through the two points to represent the linear equation.

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept.

Q: How do I find the slope of a linear equation?

A: To find the slope of a linear equation, you need to use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Q: What is the y-intercept of a linear equation?

A: The y-intercept of a linear equation is the point where the line intersects the y-axis. It is represented by the value of $b$ in the slope-intercept form of the equation.

Q: How do I use linear equations in real-world applications?

A: Linear equations have numerous real-world applications, including finance, science, and engineering. For example, you can use linear equations to calculate interest rates, model population growth, and design and optimize systems.

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

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

• Not isolating the variable on one side of the equation
• Not using the correct order of operations (PEMDAS)
• Not checking your work by plugging the values of $x$ and $y$ back into the equation

Conclusion

In conclusion, solving linear equations and finding ordered pairs that satisfy the equation is a crucial concept in mathematics. By understanding the basics of linear equations and practicing with real-world examples, you can become proficient in solving linear equations and apply them to a variety of fields.

Practice Problems

Try solving the following linear equations to practice your skills:

• $x + y = 20$
• $x - y = 5$
• $2x + 3y = 12$

Real-World Applications

Linear equations have numerous real-world applications, including:

• Finance: Linear equations are used to calculate interest rates, investment returns, and other financial metrics.
• Science: Linear equations are used to model population growth, chemical reactions, and other scientific phenomena.
• Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Tips and Tricks

• Use the order of operations (PEMDAS) to simplify the equation and make it easier to solve.
• Check your work by plugging the values of $x$ and $y$ back into the equation to ensure that they satisfy the equation.
• Use linear equations to model real-world phenomena and make predictions about future events.