Solve The Following Simultaneous Equations:${ \begin{cases} 4a + 3b = 27 \ 4a + 5b = 37 \end{cases} }${ A = \square\$} { B = \square$}$

Mar 13, 2025 by ADMIN 138 views

Solving Simultaneous Equations: A Step-by-Step Guide

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Introduction

Simultaneous 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 system of two linear equations with two variables. We will use the method of substitution and elimination to find the values of the variables.

The Problem

We are given two simultaneous equations:

{ \begin{cases} 4a + 3b = 27 \\ 4a + 5b = 37 \end{cases} \}

Our goal is to find the values of $a$ and $b$ that satisfy both equations.

Step 1: Write Down the Equations

Let's start by writing down the two equations:

{ \begin{cases} 4a + 3b = 27 \\ 4a + 5b = 37 \end{cases} \}

Step 2: Eliminate One Variable

To eliminate one variable, we can subtract the first equation from the second equation. This will give us a new equation with only one variable.

{ (4a + 5b) - (4a + 3b) = 37 - 27 \}

Simplifying the equation, we get:

{ 2b = 10 \}

Step 3: Solve for One Variable

Now that we have an equation with only one variable, we can solve for that variable. In this case, we can solve for $b$ .

{ 2b = 10 \}

Dividing both sides by 2, we get:

{ b = 5 \}

Step 4: Substitute the Value Back into One of the Original Equations

Now that we have the value of $b$ , we can substitute it back into one of the original equations to solve for the other variable. Let's use the first equation:

{ 4a + 3b = 27 \}

Substituting $b = 5$ , we get:

{ 4a + 3(5) = 27 \}

Simplifying the equation, we get:

{ 4a + 15 = 27 \}

Step 5: Solve for the Other Variable

Now that we have an equation with only one variable, we can solve for that variable. In this case, we can solve for $a$ .

{ 4a + 15 = 27 \}

Subtracting 15 from both sides, we get:

{ 4a = 12 \}

Dividing both sides by 4, we get:

{ a = 3 \}

Conclusion

In this article, we solved a system of two linear equations with two variables using the method of substitution and elimination. We found that $a = 3$ and $b = 5$ are the values that satisfy both equations.

Final Answer

The final answer is:

{ \begin{cases} a = 3 \\ b = 5 \end{cases} \}

Discussion

Solving simultaneous equations is an essential skill in mathematics, and it has numerous applications in real-life situations. In this article, we used the method of substitution and elimination to solve a system of two linear equations with two variables. We can use this method to solve more complex systems of equations, and it can be applied to various fields such as physics, engineering, and economics.

Tips and Tricks

Here are some tips and tricks to help you solve simultaneous equations:

Use the method of substitution and elimination: These two methods are the most common ways to solve simultaneous equations.
Simplify the equations: Simplifying the equations can make it easier to solve them.
Use algebraic manipulations: Algebraic manipulations such as addition, subtraction, multiplication, and division can help you solve the equations.
Check your answers: Always check your answers to make sure they satisfy both equations.

Common Mistakes

Here are some common mistakes to avoid when solving simultaneous equations:

Not simplifying the equations: Failing to simplify the equations can make it harder to solve them.
Not using algebraic manipulations: Not using algebraic manipulations can make it harder to solve the equations.
Not checking your answers: Not checking your answers can lead to incorrect solutions.

Real-Life Applications

Solving simultaneous equations has numerous real-life applications. Here are a few examples:

Physics: Solving simultaneous equations is essential in physics to describe the motion of objects.
Engineering: Solving simultaneous equations is essential in engineering to design and optimize systems.
Economics: Solving simultaneous equations is essential in economics to model and analyze economic systems.

Conclusion

In conclusion, solving simultaneous equations is an essential skill in mathematics, and it has numerous applications in real-life situations. We used the method of substitution and elimination to solve a system of two linear equations with two variables, and we found that $a = 3$ and $b = 5$ are the values that satisfy both equations. We also provided some tips and tricks to help you solve simultaneous equations and some common mistakes to avoid.

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Introduction

Solving simultaneous equations is a fundamental concept in mathematics, and it has numerous applications in real-life situations. In our previous article, we provided a step-by-step guide on how to solve a system of two linear equations with two variables. In this article, we will answer some frequently asked questions about solving simultaneous equations.

Q&A

Q: What is the difference between a system of equations and a simultaneous equation?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. A simultaneous equation is a single equation that involves two or more variables.

Q: What are the methods of solving simultaneous equations?

A: There are two main methods of solving simultaneous equations: the method of substitution and the method of elimination.

Q: What is the method of substitution?

A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation.

Q: What is the method of elimination?

A: The method of elimination involves adding or subtracting the equations to eliminate one variable and then solving for the other variable.

Q: How do I know which method to use?

A: You can use either method, but the method of elimination is often easier to use when the coefficients of the variables are the same.

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

A: Some common mistakes to avoid include not simplifying the equations, not using algebraic manipulations, and not checking your answers.

Q: How do I check my answers?

A: To check your answers, you can substitute the values of the variables back into the original equations and make sure they are true.

Q: What are some real-life applications of solving simultaneous equations?

A: Solving simultaneous equations has numerous real-life applications, including physics, engineering, and economics.

Q: Can I use a calculator to solve simultaneous equations?

A: Yes, you can use a calculator to solve simultaneous equations, but it's always a good idea to check your answers by hand.

Q: How do I solve a system of three or more equations?

A: To solve a system of three or more equations, you can use the method of substitution and elimination, or you can use a matrix method.

Q: What is a matrix method?

A: A matrix method involves representing the system of equations as a matrix and then using row operations to solve for the variables.

Q: How do I represent a system of equations as a matrix?

A: To represent a system of equations as a matrix, you can write the coefficients of the variables in the matrix and then use row operations to solve for the variables.

Conclusion

In conclusion, solving simultaneous equations is an essential skill in mathematics, and it has numerous applications in real-life situations. We provided some frequently asked questions and answers about solving simultaneous equations, and we hope this article has been helpful in clarifying some of the concepts.

Final Tips

Here are some final tips to help you solve simultaneous equations:

Practice, practice, practice: The more you practice solving simultaneous equations, the more comfortable you will become with the methods and techniques.
Use a calculator: A calculator can be a useful tool when solving simultaneous equations, but always check your answers by hand.
Check your answers: Always check your answers to make sure they satisfy both equations.
Use algebraic manipulations: Algebraic manipulations such as addition, subtraction, multiplication, and division can help you solve the equations.
Simplify the equations: Simplifying the equations can make it easier to solve them.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving simultaneous equations:

Not simplifying the equations: Failing to simplify the equations can make it harder to solve them.
Not using algebraic manipulations: Not using algebraic manipulations can make it harder to solve the equations.
Not checking your answers: Not checking your answers can lead to incorrect solutions.
Not using a calculator: Not using a calculator can make it harder to solve the equations.
Not practicing: Not practicing can make it harder to solve the equations.

Real-Life Applications

Solving simultaneous equations has numerous real-life applications, including:

Physics: Solving simultaneous equations is essential in physics to describe the motion of objects.
Engineering: Solving simultaneous equations is essential in engineering to design and optimize systems.
Economics: Solving simultaneous equations is essential in economics to model and analyze economic systems.