Solve For $x$.$5x + 9y = 9$

Introduction

In algebra, solving linear equations in two variables is a fundamental concept that helps us find the values of the variables that satisfy the equation. In this article, we will focus on solving the linear equation $5x + 9y = 9$ for the variable $x$. We will use various methods to isolate the variable $x$ and provide step-by-step solutions.

What are Linear Equations in Two Variables?

A linear equation in two variables is an equation that involves two variables, say $x$ and $y$, and is expressed in the form $ax + by = c$, where $a$, $b$, and $c$ are constants. The equation $5x + 9y = 9$ is a linear equation in two variables, where $a = 5$, $b = 9$, and $c = 9$.

Solving the Equation $5x + 9y = 9$

To solve the equation $5x + 9y = 9$ for the variable $x$, we can use the following methods:

Method 1: Isolation of the Variable $x$

We can isolate the variable $x$ by subtracting $9y$ from both sides of the equation. This gives us:

$$5x = 9 - 9y$$

Next, we can divide both sides of the equation by $5$ to get:

$$x = \frac{9 - 9y}{5}$$

This is the solution to the equation $5x + 9y = 9$ for the variable $x$.

Method 2: Using the Substitution Method

We can also use the substitution method to solve the equation $5x + 9y = 9$ for the variable $x$. Let's say we want to solve for $x$ in terms of $y$. We can start by subtracting $9y$ from both sides of the equation:

$$5x = 9 - 9y$$

Next, we can divide both sides of the equation by $5$ to get:

$$x = \frac{9 - 9y}{5}$$

This is the same solution we obtained using the first method.

Method 3: Using the Elimination Method

We can also use the elimination method to solve the equation $5x + 9y = 9$ for the variable $x$. Let's say we want to eliminate the variable $y$ by multiplying the equation by a suitable constant. We can multiply the equation by $-1$ to get:

$$-5x - 9y = -9$$

Next, we can add the two equations together to eliminate the variable $y$:

$$(5x + 9y) + (-5x - 9y) = 9 + (-9)$$

This simplifies to:

$$0 = 0$$

This is a true statement, but it doesn't help us solve for the variable $x$. However, we can use this method to eliminate the variable $y$ and solve for the variable $x$.

Method 4: Using the Graphical Method

We can also use the graphical method to solve the equation $5x + 9y = 9$ for the variable $x$. We can graph the equation on a coordinate plane and find the point of intersection between the two lines. The point of intersection represents the solution to the equation.

Conclusion

In this article, we have discussed four methods for solving the linear equation $5x + 9y = 9$ for the variable $x$. We have used the isolation method, the substitution method, the elimination method, and the graphical method to find the solution. Each method has its own advantages and disadvantages, and the choice of method depends on the specific problem and the level of difficulty.

Tips and Tricks

Here are some tips and tricks to help you solve linear equations in two variables:

• Use the isolation method: This method is the most straightforward way to solve for the variable $x$.
• Use the substitution method: This method is useful when you want to solve for one variable in terms of another variable.
• Use the elimination method: This method is useful when you want to eliminate one variable by multiplying the equation by a suitable constant.
• Use the graphical method: This method is useful when you want to visualize the solution to the equation.

Practice Problems

Here are some practice problems to help you practice solving linear equations in two variables:

• Problem 1: Solve the equation $2x + 3y = 6$ for the variable $x$.
• Problem 2: Solve the equation $4x - 2y = 10$ for the variable $x$.
• Problem 3: Solve the equation $x + 2y = 5$ for the variable $x$.

Solutions

Here are the solutions to the practice problems:

• Problem 1: $x = \frac{6 - 3y}{2}$
• Problem 2: $x = \frac{10 + 2y}{4}$
• Problem 3: $x = 5 - 2y$

Conclusion

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Introduction

In our previous article, we discussed four methods for solving linear equations in two variables: the isolation method, the substitution method, the elimination method, and the graphical method. In this article, we will answer some frequently asked questions about solving linear equations in two variables.

Q: What is a linear equation in two variables?

A: A linear equation in two variables is an equation that involves two variables, say $x$ and $y$, and is expressed in the form $ax + by = c$, where $a$, $b$, and $c$ are constants.

Q: How do I know which method to use to solve a linear equation in two variables?

A: The choice of method depends on the specific problem and the level of difficulty. If you are given a simple equation, the isolation method may be the most straightforward way to solve it. If you are given a more complex equation, the substitution method or the elimination method may be more suitable.

Q: Can I use the graphical method to solve a linear equation in two variables?

A: Yes, you can use the graphical method to solve a linear equation in two variables. This method is useful when you want to visualize the solution to the equation.

Q: How do I graph a linear equation in two variables?

A: To graph a linear equation in two variables, you can use a coordinate plane and plot the points that satisfy the equation. You can also use a graphing calculator or a computer program to graph the equation.

Q: What is the difference between the substitution method and the elimination method?

A: The substitution method involves substituting one variable in terms of another variable into the equation, while the elimination method involves eliminating one variable by multiplying the equation by a suitable constant.

Q: Can I use the substitution method to solve a linear equation in two variables if I am given a system of equations?

A: Yes, you can use the substitution method to solve a linear equation in two variables if you are given a system of equations. This method is useful when you want to solve for one variable in terms of another variable.

Q: How do I know if a linear equation in two variables has a solution?

A: A linear equation in two variables has a solution if the two lines intersect at a single point. If the two lines are parallel, the equation has no solution.

Q: Can I use a graphing calculator or a computer program to solve a linear equation in two variables?

A: Yes, you can use a graphing calculator or a computer program to solve a linear equation in two variables. These tools can help you visualize the solution to the equation and find the values of the variables that satisfy the equation.

Q: How do I check my solution to a linear equation in two variables?

A: To check your solution to a linear equation in two variables, you can plug the values of the variables into the equation and see if the equation is true.

Conclusion

In conclusion, solving linear equations in two variables is a fundamental concept in algebra that helps us find the values of the variables that satisfy the equation. We have answered some frequently asked questions about solving linear equations in two variables, including questions about the isolation method, the substitution method, the elimination method, and the graphical method. We have also provided some tips and tricks to help you solve linear equations in two variables, as well as some practice problems to help you practice your skills.

Practice Problems

Here are some practice problems to help you practice solving linear equations in two variables:

• Problem 1: Solve the equation $2x + 3y = 6$ for the variable $x$.
• Problem 2: Solve the equation $4x - 2y = 10$ for the variable $x$.
• Problem 3: Solve the equation $x + 2y = 5$ for the variable $x$.

Solutions

Here are the solutions to the practice problems:

• Problem 1: $x = \frac{6 - 3y}{2}$
• Problem 2: $x = \frac{10 + 2y}{4}$
• Problem 3: $x = 5 - 2y$

Tips and Tricks

Here are some tips and tricks to help you solve linear equations in two variables:

• Use the isolation method: This method is the most straightforward way to solve for the variable $x$.
• Use the substitution method: This method is useful when you want to solve for one variable in terms of another variable.
• Use the elimination method: This method is useful when you want to eliminate one variable by multiplying the equation by a suitable constant.
• Use the graphical method: This method is useful when you want to visualize the solution to the equation.

Conclusion

In conclusion, solving linear equations in two variables is a fundamental concept in algebra that helps us find the values of the variables that satisfy the equation. We have answered some frequently asked questions about solving linear equations in two variables, including questions about the isolation method, the substitution method, the elimination method, and the graphical method. We have also provided some tips and tricks to help you solve linear equations in two variables, as well as some practice problems to help you practice your skills.