Alex Was Asked To Solve The Following System Of Equations:$\[ 4x + 2y = 6 \\]$\[ 3x + Y = 9 \\]His Work Is Shown Below:Step 1:$\[ 3x + Y = 9 \\]$\[ y = 9 - 3x \\]Step 2:$\[ \begin{aligned} 4x + 2(9 - 3x)

Introduction

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will explore how to solve a system of equations using a step-by-step approach, with a focus on algebraic methods. We will use a specific example to illustrate the process, and provide a detailed explanation of each step.

The Problem

Alex was asked to solve the following system of equations:

${ 4x + 2y = 6 }$ ${ 3x + y = 9 }$

His work is shown below:

Step 1: Isolate y in the second equation

${ 3x + y = 9 }$ ${ y = 9 - 3x }$

Step 2: Substitute the expression for y into the first equation

${ 4x + 2(9 - 3x) = 6 }$

Discussion

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To solve the system of equations, we need to find the values of x and y that satisfy both equations simultaneously. We can do this by using the method of substitution, where we substitute the expression for y from the second equation into the first equation.

Step-by-Step Solution

Step 1: Substitute the expression for y into the first equation

${ 4x + 2(9 - 3x) = 6 }$

To substitute the expression for y, we need to multiply the term 2 by the expression (9 - 3x). This gives us:

${ 4x + 18 - 6x = 6 }$

Step 2: Simplify the equation

${ 4x + 18 - 6x = 6 }$ ${ -2x + 18 = 6 }$

Step 3: Isolate x

${ -2x + 18 = 6 }$ ${ -2x = -12 }$ ${ x = 6 }$

Step 4: Substitute the value of x into the second equation

${ 3x + y = 9 }$ ${ 3(6) + y = 9 }$ ${ 18 + y = 9 }$

Step 5: Solve for y

${ 18 + y = 9 }$ ${ y = -9 }$

Conclusion

In this article, we have shown how to solve a system of equations using a step-by-step approach. We used the method of substitution to find the values of x and y that satisfy both equations simultaneously. By following these steps, we can solve any system of equations that involves two variables.

Tips and Tricks

• When solving a system of equations, it's essential to isolate one variable in one of the equations.
• Use the method of substitution to substitute the expression for one variable into the other equation.
• Simplify the equation by combining like terms.
• Isolate the variable by adding or subtracting the same value to both sides of the equation.

Real-World Applications

Solving systems of equations has numerous real-world applications, including:

• Physics and Engineering: Systems of equations are used to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
• Economics: Systems of equations are used to model economic systems, including supply and demand, and the behavior of markets.
• Computer Science: Systems of equations are used in computer science to solve problems in computer graphics, game development, and machine learning.

Conclusion

Introduction

Solving systems of equations is a fundamental concept in mathematics that can be a bit challenging for some students. In this article, we will answer some of the most frequently asked questions about solving systems of equations, providing a clear and concise explanation of each concept.

Q: What is a system of equations?

A system of equations is a set of two or more equations that involve two or more variables. For example:

${ 4x + 2y = 6 }$ ${ 3x + y = 9 }$

Q: How do I solve a system of equations?

There are several methods to solve a system of equations, including:

• Substitution method: This involves substituting the expression for one variable into the other equation.
• Elimination method: This involves adding or subtracting the same value to both sides of the equation to eliminate one variable.
• Graphical method: This involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

The substitution method involves substituting the expression for one variable into the other equation. For example:

${ 4x + 2y = 6 }$ ${ y = 9 - 3x }$

Substituting the expression for y into the first equation gives us:

${ 4x + 2(9 - 3x) = 6 }$

Q: What is the elimination method?

The elimination method involves adding or subtracting the same value to both sides of the equation to eliminate one variable. For example:

${ 4x + 2y = 6 }$ ${ 3x + y = 9 }$

Multiplying the second equation by 2 gives us:

${ 6x + 2y = 18 }$

Subtracting the first equation from the second equation gives us:

${ 2x = 12 }$

Q: How do I know which method to use?

The choice of method depends on the specific system of equations. If the equations are linear and have the same slope, the substitution method may be the best choice. If the equations are linear and have different slopes, the elimination method may be the best choice.

Q: What are some common mistakes to avoid?

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

• Not isolating one variable: Make sure to isolate one variable in one of the equations before substituting or eliminating.
• Not simplifying the equation: Make sure to simplify the equation by combining like terms.
• Not checking the solution: Make sure to check the solution by substituting the values back into the original equations.

Q: How do I check my solution?

To check your solution, substitute the values back into the original equations. If the values satisfy both equations, then the solution is correct.

Conclusion

Solving systems of equations can be a bit challenging, but with practice and patience, you can master this skill. By understanding the different methods and common mistakes to avoid, you can become more confident in your ability to solve systems of equations. Whether you're a student or a professional, solving systems of equations is an essential skill that can be applied to a wide range of problems.