Linear simultaneous equations

Oct 13, 2025 by windailys.com 24 views

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 explore the process of solving simultaneous equations, using the given example of two linear equations in two variables. We will break down the solution into manageable steps, making it easy to understand and apply.

What are Simultaneous Equations?

Simultaneous equations are a set of two or more equations that involve two or more variables. In other words, they are equations that are solved together to find the values of the variables. The equations are said to be simultaneous because they are solved at the same time, rather than separately.

The Given Equations

The given equations are:

{ \begin{array}{l} 2x + 4y = 1 \\ 3x - 5y = 7 \end{array} \}

These equations are linear, meaning that they are in the form of a straight line. The first equation is in the form of $2x + 4y = 1$ , and the second equation is in the form of $3x - 5y = 7$ .

Step 1: Write Down the Equations

The first step in solving simultaneous equations is to write down the equations. In this case, we have:

{ \begin{array}{l} 2x + 4y = 1 \\ 3x - 5y = 7 \end{array} \}

Step 2: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the equations by necessary multiples. Let's multiply the first equation by 3 and the second equation by 2.

{ \begin{array}{l} 6x + 12y = 3 \\ 6x - 10y = 14 \end{array} \}

Step 3: Subtract the Second Equation from the First Equation

Now, let's subtract the second equation from the first equation to eliminate the variable x.

{ (6x + 12y) - (6x - 10y) = 3 - 14 \}

Simplifying the equation, we get:

{ 22y = -11 \}

Step 4: Solve for y

Now, let's solve for y by dividing both sides of the equation by 22.

{ y = \frac{-11}{22} \}

Simplifying the equation, we get:

{ y = -\frac{1}{2} \}

Step 5: Substitute the Value of y into One of the Original Equations

Now that we have the value of y, let's substitute it into one of the original equations to solve for x. Let's use the first equation.

{ 2x + 4\left(-\frac{1}{2}\right) = 1 \}

Simplifying the equation, we get:

{ 2x - 2 = 1 \}

Adding 2 to both sides of the equation, we get:

{ 2x = 3 \}

Dividing both sides of the equation by 2, we get:

{ x = \frac{3}{2} \}

Conclusion

In this article, we have solved the simultaneous equations using the given example of two linear equations in two variables. We have broken down the solution into manageable steps, making it easy to understand and apply. The steps involved in solving simultaneous equations are:

Write down the equations
Multiply the equations by necessary multiples
Subtract the second equation from the first equation
Solve for y
Substitute the value of y into one of the original equations to solve for x

By following these steps, you can solve simultaneous equations with ease.

Example Problems

Here are a few example problems to help you practice solving simultaneous equations.

Example 1

Solve the simultaneous equations:

{ \begin{array}{l} x + 2y = 6 \\ 3x - 2y = 2 \end{array} \}

Solution

To solve the simultaneous equations, we need to follow the steps outlined above.

Write down the equations:

{ \begin{array}{l} x + 2y = 6 \\ 3x - 2y = 2 \end{array} \}

Multiply the equations by necessary multiples:

{ \begin{array}{l} x + 2y = 6 \\ 9x - 6y = 6 \end{array} \}

Subtract the second equation from the first equation:

{ (x + 2y) - (9x - 6y) = 6 - 6 \}

Simplifying the equation, we get:

{ -8x + 8y = 0 \}

Solve for y:

{ y = \frac{8x}{8} \}

Simplifying the equation, we get:

{ y = x \}

Substitute the value of y into one of the original equations to solve for x:

{ x + 2(x) = 6 \}

Simplifying the equation, we get:

{ 3x = 6 \}

Dividing both sides of the equation by 3, we get:

{ x = 2 \}

Example 2

Solve the simultaneous equations:

{ \begin{array}{l} 2x + 3y = 12 \\ x - 2y = -3 \end{array} \}

Solution

To solve the simultaneous equations, we need to follow the steps outlined above.

Write down the equations:

{ \begin{array}{l} 2x + 3y = 12 \\ x - 2y = -3 \end{array} \}

Multiply the equations by necessary multiples:

{ \begin{array}{l} 4x + 6y = 24 \\ x - 2y = -3 \end{array} \}

Subtract the second equation from the first equation:

{ (4x + 6y) - (x - 2y) = 24 - (-3) \}

Simplifying the equation, we get:

{ 3x + 8y = 27 \}

Solve for y:

{ y = \frac{27 - 3x}{8} \}

Substitute the value of y into one of the original equations to solve for x:

{ 2x + 3\left(\frac{27 - 3x}{8}\right) = 12 \}

Simplifying the equation, we get:

{ 16x + 81 - 9x = 96 \}

Combining like terms, we get:

{ 7x = 15 \}

Dividing both sides of the equation by 7, we get:

{ x = \frac{15}{7} \}

Example 3

Solve the simultaneous equations:

{ \begin{array}{l} x + 2y = 8 \\ 2x + 3y = 11 \end{array} \}

Solution

To solve the simultaneous equations, we need to follow the steps outlined above.

Write down the equations:

{ \begin{array}{l} x + 2y = 8 \\ 2x + 3y = 11 \end{array} \}

Multiply the equations by necessary multiples:

{ \begin{array}{l} x + 2y = 8 \\ 4x + 6y = 22 \end{array} \}

Subtract the second equation from the first equation:

{ (x + 2y) - (4x + 6y) = 8 - 22 \}

Simplifying the equation, we get:

{ -3x - 4y = -14 \}

Solve for y:

{ y = \frac{-14 + 3x}{4} \}

Substitute the value of y into one of the original equations to solve for x:

{ x + 2\left(\frac{-14 + 3x}{4}\right) = 8 \}

Simplifying the equation, we get:

{ x - \frac{28}{2} + \frac{3x}{2} = 8 \}

Combining like terms, we get:

{ \frac{5x}{2} = \frac{36}{2} \}

Dividing both sides of the equation by 5/2, we get:

{ x = \frac{36}{5/2} \}

Simplifying the equation, we get:

{ x = \frac{72}{5} \}

Conclusion

Write down the equations
Frequently Asked Questions (FAQs) About Solving Simultaneous Equations ====================================================================

Q: What are simultaneous equations?

A: Simultaneous equations are a set of two or more equations that involve two or more variables. In other words, they are equations that are solved together to find the values of the variables.

Q: How do I know if I have a system of simultaneous equations?

A: A system of simultaneous equations is a set of two or more equations that involve two or more variables. If you have a set of equations that involve two or more variables, and you need to solve them together, then you have a system of simultaneous equations.

Q: What are the different types of simultaneous equations?

A: There are two main types of simultaneous equations:

Linear simultaneous equations: These are equations that are in the form of a straight line. They can be solved using the methods outlined in this article.
Non-linear simultaneous equations: These are equations that are not in the form of a straight line. They can be solved using more advanced methods, such as substitution and elimination.

Q: How do I solve simultaneous equations?

A: To solve simultaneous equations, you need to follow the steps outlined in this article:

Write down the equations
Multiply the equations by necessary multiples
Subtract the second equation from the first equation
Solve for y
Substitute the value of y into one of the original equations to solve for x

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

A: If you have a system of three or more simultaneous equations, you can use the same methods outlined in this article to solve them. However, you may need to use more advanced methods, such as substitution and elimination, to solve the system.

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

A: Yes, you can use a calculator to solve simultaneous equations. However, it's always a good idea to check your work by hand to make sure that the solution is correct.

Q: What if I get stuck while solving simultaneous equations?

A: If you get stuck while solving simultaneous equations, don't worry! You can try the following:

Go back and review the steps outlined in this article
Check your work for errors
Ask a teacher or tutor for help
Use a calculator to check your work

Q: Are simultaneous equations used in real-life situations?

A: Yes, simultaneous equations are used in many real-life situations, such as:

Physics: Simultaneous equations are used to solve problems involving motion, forces, and energy.
Engineering: Simultaneous equations are used to solve problems involving design, construction, and operation of systems.
Economics: Simultaneous equations are used to solve problems involving supply and demand, prices, and production.
Computer Science: Simultaneous equations are used to solve problems involving algorithms, data structures, and computer networks.

Q: Can I use simultaneous equations to solve problems involving more than two variables?

A: Yes, you can use simultaneous equations to solve problems involving more than two variables. However, you may need to use more advanced methods, such as substitution and elimination, to solve the system.

Q: Are there any online resources that can help me learn more about solving simultaneous equations?

A: Yes, there are many online resources that can help you learn more about solving simultaneous equations, such as:

Khan Academy: Khan Academy has a comprehensive video series on solving simultaneous equations.
Mathway: Mathway is an online calculator that can help you solve simultaneous equations.
Wolfram Alpha: Wolfram Alpha is an online calculator that can help you solve simultaneous equations.
Math Open Reference: Math Open Reference is an online reference book that can help you learn more about solving simultaneous equations.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving simultaneous equations. We have also provided some tips and resources to help you learn more about solving simultaneous equations. Whether you are a student or a professional, solving simultaneous equations is an essential skill that can help you solve problems in many different fields.