Solve The Equation: 4 Y + 3 X = 40 4y + 3x = 40 4 Y + 3 X = 40

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a simple linear equation, $4y + 3x = 40$, using various methods and techniques. We will break down the solution into manageable steps, making it easy to understand and follow.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its components. The equation is in the form of $ax + by = c$, where $a$, $b$, and $c$ are constants. In this case, $a = 3$, $b = 4$, and $c = 40$. The variables $x$ and $y$ are the unknowns that we need to solve for.

Method 1: Substitution Method

One way to solve the equation is by using the substitution method. This method involves isolating one variable in terms of the other and then substituting it into the original equation.

Step 1: Isolate One Variable

Let's isolate the variable $y$ in terms of $x$. We can do this by subtracting $3x$ from both sides of the equation:

$$4y = 40 - 3x$$

Step 2: Solve for $y$

Now that we have isolated $y$, we can solve for it by dividing both sides of the equation by 4:

$$y = \frac{40 - 3x}{4}$$

Step 3: Substitute $y$ into the Original Equation

Now that we have expressed $y$ in terms of $x$, we can substitute it into the original equation:

$$4\left(\frac{40 - 3x}{4}\right) + 3x = 40$$

Step 4: Simplify the Equation

Simplifying the equation, we get:

$$40 - 3x + 3x = 40$$

Step 5: Solve for $x$

Since the equation simplifies to $40 = 40$, we can conclude that the equation has infinitely many solutions. However, we can still find a specific solution by setting $x = 0$.

Method 2: Elimination Method

Another way to solve the equation is by using the elimination method. This method involves multiplying the equation by a constant to eliminate one of the variables.

Step 4: Multiply the Equation by a Constant

Let's multiply the equation by 4 to eliminate the variable $y$:

$$16y + 12x = 160$$

Step 5: Subtract the Original Equation

Now that we have multiplied the equation by 4, we can subtract the original equation to eliminate the variable $y$:

$$(16y + 12x) - (4y + 3x) = 160 - 40$$

Step 6: Simplify the Equation

Simplifying the equation, we get:

$$12y + 9x = 120$$

Step 7: Solve for $x$

Now that we have eliminated the variable $y$, we can solve for $x$ by dividing both sides of the equation by 9:

$$x = \frac{120}{9}$$

Step 8: Solve for $y$

Now that we have found the value of $x$, we can substitute it into the original equation to find the value of $y$:

$$4y + 3\left(\frac{120}{9}\right) = 40$$

Step 9: Simplify the Equation

Simplifying the equation, we get:

$$4y + \frac{360}{9} = 40$$

Step 10: Solve for $y$

Now that we have simplified the equation, we can solve for $y$ by subtracting $\frac{360}{9}$ from both sides of the equation:

$$4y = 40 - \frac{360}{9}$$

Step 11: Simplify the Equation

Simplifying the equation, we get:

$$4y = \frac{360 - 360}{9}$$

Step 12: Solve for $y$

Since the equation simplifies to $4y = 0$, we can conclude that $y = 0$.

Conclusion

$$$$$$

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form of $ax + by = c$, where $a$, $b$, and $c$ are constants.

Q: What are the different methods for solving linear equations?

A: There are several methods for solving linear equations, including:

• Substitution method
• Elimination method
• Graphical method
• Algebraic method

Q: What is the substitution method?

A: The substitution method is a method for solving linear equations in which one variable is expressed in terms of the other variable, and then substituted into the original equation.

Q: What is the elimination method?

A: The elimination method is a method for solving linear equations in which one of the variables is eliminated by multiplying the equation by a constant, and then subtracting the original equation.

Q: How do I know which method to use?

A: The choice of method depends on the specific equation and the variables involved. If the equation has two variables, the substitution method may be more convenient. If the equation has multiple variables, the elimination method may be more efficient.

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

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

• Not checking the solution in the original equation
• Not simplifying the equation before solving
• Not using the correct method for the specific equation
• Not checking for extraneous solutions

Q: How do I check my solution?

A: To check your solution, substitute the values of the variables back into the original equation and simplify. If the equation is true, then the solution is correct.

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

A: Linear equations have many real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects under constant acceleration.
• Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Linear equations are used to model economic systems and make predictions about future trends.
• Computer Science: Linear equations are used in algorithms and data structures to solve problems efficiently.

Q: Can linear equations be used to solve systems of equations?

A: Yes, linear equations can be used to solve systems of equations. A system of equations is a set of two or more linear equations that are solved simultaneously.

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

A: To solve a system of linear equations, use the substitution method or the elimination method to solve one of the equations for one of the variables, and then substitute that value into the other equation.

Q: What are some common types of systems of linear equations?

A: Some common types of systems of linear equations include:

• Homogeneous systems: A system of linear equations in which all the constants are zero.
• Non-homogeneous systems: A system of linear equations in which not all the constants are zero.
• Consistent systems: A system of linear equations in which the solution is unique.
• Inconsistent systems: A system of linear equations in which there is no solution.

Q: How do I determine if a system of linear equations is consistent or inconsistent?

A: To determine if a system of linear equations is consistent or inconsistent, use the following steps:

• Write the augmented matrix for the system.
• Perform row operations to put the matrix in row-echelon form.
• If the matrix has a row of zeros, the system is inconsistent.
• If the matrix has no row of zeros, the system is consistent.

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

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

• Not checking the solution in the original equations
• Not simplifying the equations before solving
• Not using the correct method for the specific system
• Not checking for extraneous solutions