Find The Solution To This System:Equation 1: 5 X − 2 Y = − 11 5x - 2y = -11 5 X − 2 Y = − 11 Equation 2: − 2 X + 5 Y = 17 -2x + 5y = 17 − 2 X + 5 Y = 17 Step 1: To Create X X X -coefficients That Are Additive Inverses, Equation 1 Can Be Multiplied By □ \square □ .Multiplying Equation 2

Mar 9, 2025 by ADMIN 287 views

**Solving a System of Linear Equations: A Step-by-Step Guide**

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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, economics, and computer science. In this article, we will focus on solving a system of two linear equations using the method of substitution and elimination. We will use the given system of equations as an example to illustrate the steps involved in solving it.

The System of Equations

The system of equations we will be solving is:

Equation 1: $5x - 2y = -11$ Equation 2: $-2x + 5y = 17$

Step 1: Multiply Equation 1 by a Constant

To create $x$ -coefficients that are additive inverses, we need to multiply Equation 1 by a constant. Let's multiply Equation 1 by 2.

# Multiply Equation 1 by 2
equation1_multiplied = 2 * (5*x - 2*y)
print(equation1_multiplied)

This will give us:

$10x - 4y = -22$

Step 2: Multiply Equation 2 by a Constant

To create $x$ -coefficients that are additive inverses, we need to multiply Equation 2 by 5.

# Multiply Equation 2 by 5
equation2_multiplied = 5 * (-2*x + 5*y)
print(equation2_multiplied)

This will give us:

$-10x + 25y = 85$

Step 3: Add the Two Equations

Now that we have the two equations with additive inverse $x$ -coefficients, we can add them to eliminate the $x$ -variable.

# Add the two equations
added_equations = (10*x - 4*y) + (-10*x + 25*y)
print(added_equations)

This will give us:

$21y = 63$

Step 4: Solve for y

Now that we have the equation $21y = 63$ , we can solve for $y$ by dividing both sides by 21.

# Solve for y
y = 63 / 21
print(y)

This will give us:

$y = 3$

Step 5: Substitute y into One of the Original Equations

Now that we have the value of $y$ , we can substitute it into one of the original equations to solve for $x$ . Let's substitute $y = 3$ into Equation 1.

# Substitute y into Equation 1
equation1_substituted = 5*x - 2*(3)
print(equation1_substituted)

This will give us:

$5x - 6 = -11$

Step 6: Solve for x

Now that we have the equation $5x - 6 = -11$ , we can solve for $x$ by adding 6 to both sides and then dividing both sides by 5.

# Solve for x
x = (-11 + 6) / 5
print(x)

This will give us:

$x = -1$

Conclusion

In this article, we have solved a system of two linear equations using the method of substitution and elimination. We have shown that by multiplying the equations by appropriate constants, we can create $x$ -coefficients that are additive inverses, allowing us to eliminate the $x$ -variable and solve for $y$ . We have then substituted the value of $y$ into one of the original equations to solve for $x$ . The final solution to the system of equations is $x = -1$ and $y = 3$ .

Discussion

Solving a system of linear equations is an important concept in mathematics, and it has numerous applications in various fields. The method of substitution and elimination is a powerful tool for solving systems of linear equations, and it can be used to solve systems with any number of equations and variables. In this article, we have shown that by following the steps outlined above, we can solve a system of two linear equations using the method of substitution and elimination.

Example Problems

Here are some example problems that you can try to practice solving systems of linear equations using the method of substitution and elimination.

Solve the system of equations: $2x + 3y = 7$ $x - 2y = -3$
Solve the system of equations: $x + 2y = 4$ $3x - 2y = 5$
Solve the system of equations: $4x - 3y = 2$ $2x + 5y = 11$

Tips and Tricks

Here are some tips and tricks that you can use to help you solve systems of linear equations using the method of substitution and elimination.

Make sure to multiply the equations by the correct constants to create $x$ -coefficients that are additive inverses.
Make sure to add the equations correctly to eliminate the $x$ -variable.
Make sure to substitute the value of $y$ into one of the original equations to solve for $x$ .
Make sure to check your work by plugging the values of $x$ and $y$ back into the original equations to make sure they are true.

Conclusion

Solving a system of linear equations is an important concept in mathematics, and it has numerous applications in various fields. The method of substitution and elimination is a powerful tool for solving systems of linear equations, and it can be used to solve systems with any number of equations and variables. By following the steps outlined above, you can solve a system of two linear equations using the method of substitution and elimination.

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Q: What is a system of linear equations?

A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.

Q: What is the method of substitution and elimination?

The method of substitution and elimination is a technique used to solve systems of linear equations. It involves substituting the value of one variable into one of the equations to solve for the other variable.

Q: How do I know which method to use?

You can use either the method of substitution or elimination to solve a system of linear equations. The choice of method depends on the specific system of equations and the values of the variables.

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

The method of substitution involves substituting the value of one variable into one of the equations to solve for the other variable. The method of elimination involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I multiply the equations by the correct constants?

To multiply the equations by the correct constants, you need to multiply the coefficients of the variables by the same constant. For example, if you have the equation 2x + 3y = 7, you can multiply it by 2 to get 4x + 6y = 14.

Q: How do I add the equations correctly to eliminate the x-variable?

To add the equations correctly, you need to add the coefficients of the x-variable and the constant terms. For example, if you have the equations 2x + 3y = 7 and x - 2y = -3, you can add them to get 3x + y = 4.

Q: How do I substitute the value of y into one of the original equations to solve for x?

To substitute the value of y into one of the original equations, you need to replace the variable y with its value. For example, if you have the equation 2x + 3y = 7 and y = 2, you can substitute y = 2 into the equation to get 2x + 6 = 7.

Q: How do I check my work by plugging the values of x and y back into the original equations?

To check your work, you need to plug the values of x and y back into the original equations to make sure they are true. For example, if you have the equations 2x + 3y = 7 and x - 2y = -3, you can plug x = 2 and y = 2 into the equations to make sure they are true.

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

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

Not multiplying the equations by the correct constants
Not adding the equations correctly to eliminate the x-variable
Not substituting the value of y into one of the original equations to solve for x
Not checking your work by plugging the values of x and y back into the original equations

Q: How do I use the method of substitution and elimination to solve systems of linear equations with more than two variables?

To use the method of substitution and elimination to solve systems of linear equations with more than two variables, you need to follow the same steps as before, but with more variables. You can use the method of substitution to substitute the value of one variable into one of the equations to solve for another variable, and then use the method of elimination to eliminate one of the variables.

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

Some real-world applications of solving systems of linear equations include:

Physics: Solving systems of linear equations is used to model the motion of objects and to solve problems involving forces and energies.
Engineering: Solving systems of linear equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Solving systems of linear equations is used to model economic systems and to solve problems involving supply and demand.
Computer Science: Solving systems of linear equations is used in computer graphics and game development to create 3D models and to solve problems involving physics and motion.

Q: How do I use technology to solve systems of linear equations?

You can use technology, such as calculators and computer software, to solve systems of linear equations. Some popular software for solving systems of linear equations includes:

MATLAB
Mathematica
Python
R

Q: What are some tips for solving systems of linear equations?

Some tips for solving systems of linear equations include:

Make sure to read the problem carefully and understand what is being asked.
Use the method of substitution and elimination to solve the system of equations.
Check your work by plugging the values of x and y back into the original equations.
Use technology, such as calculators and computer software, to solve the system of equations.
Practice solving systems of linear equations to become more comfortable with the method.