If $x = 2y + 3$ And $4x - 5y = 9$, What Is The Value Of $y$?

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Introduction

Solving a system of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, and economics. In this article, we will focus on solving a system of two linear equations with two variables, $x$ and $y$. We will use the given equations, $x = 2y + 3$ and $4x - 5y = 9$, to find the value of $y$.

Understanding the Equations

The first equation, $x = 2y + 3$, represents a linear relationship between $x$ and $y$. This equation can be rewritten as $x - 2y = 3$, which is in the slope-intercept form, $y = mx + b$. The slope of this line is $-2$, and the y-intercept is $3$.

The second equation, $4x - 5y = 9$, represents another linear relationship between $x$ and $y$. This equation can be rewritten as $4x - 5y - 9 = 0$, which is in the standard form, $Ax + By = C$. The coefficients of this equation are $A = 4$, $B = -5$, and $C = -9$.

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the method of substitution.

First, we will solve the first equation for $x$:

$x = 2y + 3$

Next, we will substitute this expression for $x$ into the second equation:

$4(2y + 3) - 5y = 9$

Expanding and simplifying the equation, we get:

$8y + 12 - 5y = 9$

Combine like terms:

$3y + 12 = 9$

Subtract 12 from both sides:

$3y = -3$

Divide both sides by 3:

$y = -1$

Conclusion

In this article, we solved a system of two linear equations with two variables, $x$ and $y$. We used the method of substitution to find the value of $y$. The final answer is $y = -1$.

Final Answer

The final answer is $\boxed{-1}$.

Related Topics

• Solving systems of linear equations
• Linear equations
• Algebra
• Mathematics

References

• [1] "Linear Equations" by Khan Academy
• [2] "Systems of Linear Equations" by Mathway
• [3] "Algebra" by Wikipedia

Further Reading

• "Solving Systems of Linear Equations" by MIT OpenCourseWare
• "Linear Algebra" by Stanford University
• "Algebra and Trigonometry" by Paul's Online Math Notes
  

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, and economics. In our previous article, we solved a system of two linear equations with two variables, $x$ and $y$. In this article, we will answer some frequently asked questions about solving 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 involve two or more variables. 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 several methods to solve a system of linear equations, including the method of substitution, the method of elimination, and the method of matrices. The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one variable. The method of matrices involves using matrices to represent the system of equations and then solving for the variables.

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

A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one variable. The method of substitution is often used when one of the equations is easy to solve for one variable, while the method of elimination is often used when the equations are more complex.

Q: How do I know which method to use?

A: The choice of method depends on the specific system of equations and the variables involved. If one of the equations is easy to solve for one variable, the method of substitution may be the best choice. If the equations are more complex, the method of elimination may be the best choice.

Q: What if I have a system of three or more linear equations?

A: If you have a system of three or more linear equations, you can use the method of matrices to solve the system. This involves representing the system of equations as a matrix and then using row operations to solve for the variables.

Q: What if I have a system of linear equations with no solution?

A: If you have a system of linear equations with no solution, it means that the equations are inconsistent. This can happen if the equations are contradictory, such as $x = 2$ and $x = 3$. In this case, there is no value of $x$ that can satisfy both equations.

Q: What if I have a system of linear equations with infinitely many solutions?

A: If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent. This can happen if one of the equations is a multiple of the other equation, such as $x = 2y$ and $2x = 4y$. In this case, there are infinitely many values of $x$ and $y$ that can satisfy both equations.

Q: How do I graph a system of linear equations?

A: To graph a system of linear equations, you can use a graphing calculator or a computer program to plot the equations on a coordinate plane. You can then identify the point of intersection, which represents the solution to the system.

Q: What are some real-world applications of solving systems of linear equations?

A: Solving systems of linear equations has numerous applications in various fields such as physics, engineering, and economics. Some examples include:

• Finding the intersection of two lines in a coordinate plane
• Determining the cost of producing a product
• Calculating the interest rate on a loan
• Finding the maximum or minimum value of a function

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields. In this article, we answered some frequently asked questions about solving systems of linear equations, including the method of substitution, the method of elimination, and the method of matrices. We also discussed some real-world applications of solving systems of linear equations.

Final Answer

The final answer is $\boxed{0}$.

Related Topics

• Solving systems of linear equations
• Linear equations
• Algebra
• Mathematics

References

• [1] "Linear Equations" by Khan Academy
• [2] "Systems of Linear Equations" by Mathway
• [3] "Algebra" by Wikipedia

Further Reading

• "Solving Systems of Linear Equations" by MIT OpenCourseWare
• "Linear Algebra" by Stanford University
• "Algebra and Trigonometry" by Paul's Online Math Notes