Which Ordered Pair Is A Solution Of The Equation 2 X − Y = 9 2x - Y = 9 2 X − Y = 9 ?A. ( − 4 , 1 (-4, 1 ( − 4 , 1 ] B. ( − 2 , 5 (-2, 5 ( − 2 , 5 ] C. ( 5 , 1 (5, 1 ( 5 , 1 ] D. ( 6 , − 3 (6, -3 ( 6 , − 3 ]

Mar 9, 2025 by ADMIN 211 views

**Solving Linear Equations: A Step-by-Step Guide**

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill to master. In this article, we will focus on solving a specific linear equation, $2x - y = 9$ , and determine which ordered pair is a solution to this equation.

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form $ax + by = c$ , where $a$ , $b$ , and $c$ are constants, and $x$ and $y$ are variables. Linear equations can be solved using various methods, including substitution, elimination, and graphing.

The Equation $2x - y = 9$

The given equation is $2x - y = 9$ . To solve this equation, we need to isolate the variable $y$ . We can do this by adding $y$ to both sides of the equation, which gives us $2x = y + 9$ . Next, we can subtract 9 from both sides to get $2x - 9 = y$ .

Substitution Method

One way to solve the equation $2x - y = 9$ is to use the substitution method. We can substitute the expression for $y$ from the previous step into the original equation. This gives us $2x - (2x - 9) = 9$ . Simplifying this equation, we get $9 = 9$ , which is a true statement. This means that the equation $2x - y = 9$ is an identity, and any ordered pair that satisfies the equation will be a solution.

Ordered Pairs

Now that we have determined that the equation $2x - y = 9$ is an identity, we can examine the given ordered pairs to see which one satisfies the equation. The ordered pairs are:

A. $(-4, 1)$
B. $(-2, 5)$
C. $(5, 1)$
D. $(6, -3)$

Checking the Ordered Pairs

To determine which ordered pair is a solution to the equation $2x - y = 9$ , we can substitute each pair into the equation and check if it is true. Let's start with pair A.

For pair A, $x = -4$ and $y = 1$ . Substituting these values into the equation, we get $2(-4) - 1 = 9$ , which simplifies to $-8 - 1 = 9$ . This is not true, so pair A is not a solution.

Next, let's check pair B.

For pair B, $x = -2$ and $y = 5$ . Substituting these values into the equation, we get $2(-2) - 5 = 9$ , which simplifies to $-4 - 5 = 9$ . This is not true, so pair B is not a solution.

Now, let's check pair C.

For pair C, $x = 5$ and $y = 1$ . Substituting these values into the equation, we get $2(5) - 1 = 9$ , which simplifies to $10 - 1 = 9$ . This is true, so pair C is a solution.

Finally, let's check pair D.

For pair D, $x = 6$ and $y = -3$ . Substituting these values into the equation, we get $2(6) - (-3) = 9$ , which simplifies to $12 + 3 = 9$ . This is not true, so pair D is not a solution.

Conclusion

In conclusion, the ordered pair that is a solution to the equation $2x - y = 9$ is pair C, $(5, 1)$ . This is because when we substitute the values $x = 5$ and $y = 1$ into the equation, we get a true statement. The other ordered pairs, A, B, and D, are not solutions to the equation.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations and to check your work by plugging in the values of the variables into the equation. Additionally, make sure to simplify the equation as much as possible to make it easier to solve.

Common Mistakes

One common mistake when solving linear equations is to forget to check the work. This can lead to incorrect solutions. Another mistake is to not simplify the equation enough, which can make it more difficult to solve.

Real-World Applications

Linear equations have many real-world applications, including physics, engineering, and economics. For example, the equation $2x - y = 9$ can be used to model a situation where the cost of producing a certain product is related to the number of units produced and the price per unit.

Final Thoughts

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. 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 linear equation?

A: There are several methods to solve a linear equation, including substitution, elimination, and graphing. The method you choose will depend on the specific equation and the variables involved.

Q: What is the substitution method?

A: The substitution method involves substituting the expression for one variable into the equation in terms of the other variable. This can be done by solving one of the equations for one variable and then substituting that expression into the other equation.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one of the variables. This can be done by multiplying one or both of the equations by a constant to make the coefficients of one of the variables the same.

Q: How do I graph a linear equation?

A: To graph a linear equation, you can use the slope-intercept form of the equation, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. You can also use the point-slope form of the equation, which is $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line.

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

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, the equation $2x - y = 9$ is a linear equation, while the equation $x^2 + 4x + 4 = 0$ is a quadratic equation.

Q: Can I solve a linear equation with more than two variables?

A: Yes, you can solve a linear equation with more than two variables. However, the process can be more complex and may involve using matrices or other advanced techniques.

Q: How do I check my work when solving a linear equation?

A: To check your work, you can plug in the values of the variables into the equation and see if it is true. You can also use a calculator or a computer program to check your work.

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

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

Forgetting to check your work
Not simplifying the equation enough
Making errors when multiplying or dividing
Not using the correct method for the specific equation

Q: How do I apply linear equations to real-world problems?

A: Linear equations can be used to model a wide range of real-world problems, including physics, engineering, and economics. For example, the equation $2x - y = 9$ can be used to model a situation where the cost of producing a certain product is related to the number of units produced and the price per unit.

Q: What are some advanced topics in linear equations?

A: Some advanced topics in linear equations include:

Systems of linear equations
Matrices
Determinants
Linear transformations

Q: How do I learn more about linear equations?

A: There are many resources available to learn more about linear equations, including textbooks, online tutorials, and video lectures. You can also practice solving linear equations by working through problems and exercises.