Multiply the First Equation by 10 and the Second Equation by 9

Oct 13, 2025 by windailys.com 51 views

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Introduction

In this article, we will explore the method of elimination to solve a system of linear equations. The system of linear equations is a set of two or more equations that involve two or more variables. The method of elimination is a popular technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables.

What is the Method of Elimination?

The method of elimination is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. This method involves multiplying one or both equations by a constant to make the coefficients of one of the variables the same, but with opposite signs. Once the coefficients are the same, but with opposite signs, we can add or subtract the equations to eliminate one of the variables.

Step 1: Multiply the Equations by Necessary Multiples

To solve the system of linear equations using the method of elimination, we need to multiply the equations by necessary multiples to make the coefficients of one of the variables the same, but with opposite signs.

Multiply the First Equation by 10 and the Second Equation by 9

To eliminate the variable y, we need to make the coefficients of y the same, but with opposite signs. We can do this by multiplying the first equation by 10 and the second equation by 9.

# Define the coefficients of the equations
a1 = 5
b1 = 9
c1 = -10
a2 = 7
b2 = 10
c2 = -1
a1_new = a1 * 10
b1_new = b1 * 10
c1_new = c1 * 10
a2_new = a2 * 9
b2_new = b2 * 9
c2_new = c2 * 9

Multiply the First Equation by 10 and the Second Equation by 9

After multiplying the equations, we get:

# Print the new equations
print("10(5x + 9y = -10)")
print("9(7x + 10y = -1)")

Step 2: Add or Subtract the Equations

Now that we have the new equations, we can add or subtract them to eliminate one of the variables.

Add the New Equations

To eliminate the variable y, we can add the new equations.

# Add the new equations
a_new = a1_new + a2_new
b_new = b1_new + b2_new
c_new = c1_new + c2_new

Add the New Equations

After adding the new equations, we get:

# Print the new equation
print("10(5x + 9y = -10) + 9(7x + 10y = -1)")

Step 3: Solve for the Variable

Now that we have the new equation, we can solve for the variable x.

Solve for x

To solve for x, we can divide both sides of the equation by the coefficient of x.

# Solve for x
x = c_new / a_new

Solve for x

After solving for x, we get:

# Print the value of x
print("x =", x)

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

Now that we have the value of x, we can substitute it into one of the original equations to solve for the variable y.

Substitute the Value of x into the First Equation

To solve for y, we can substitute the value of x into the first equation.

# Substitute the value of x into the first equation
y = (c1 - a1 * x) / b1

Substitute the Value of x into the First Equation

After substituting the value of x into the first equation, we get:

# Print the value of y
print("y =", y)

Conclusion

In this article, we have explored the method of elimination to solve a system of linear equations. We have shown how to multiply the equations by necessary multiples to make the coefficients of one of the variables the same, but with opposite signs. We have also shown how to add or subtract the equations to eliminate one of the variables. Finally, we have shown how to solve for the variables x and y.

Example Use Case

The method of elimination can be used to solve a wide range of systems of linear equations. For example, we can use this method to solve the system of linear equations:

# Define the coefficients of the equations
a1 = 2
b1 = 3
c1 = 5
a2 = 4
b2 = 5
c2 = 6
a1_new = a1 * 5
b1_new = b1 * 5
c1_new = c1 * 5
a2_new = a2 * 3
b2_new = b2 * 3
c2_new = c2 * 3
a_new = a1_new + a2_new
b_new = b1_new + b2_new
c_new = c1_new + c2_new
x = c_new / a_new
y = (c1 - a1 * x) / b1
print("x =", x)
print("y =", y)

This code will output the values of x and y that satisfy the system of linear equations.

Code Implementation

The code implementation for the method of elimination can be implemented in a variety of programming languages, including Python, Java, and C++. Here is an example implementation in Python:

def solve_system(a1, b1, c1, a2, b2, c2):
    # Multiply the first equation by 10 and the second equation by 9
    a1_new = a1 * 10
    b1_new = b1 * 10
    c1_new = c1 * 10
    a2_new = a2 * 9
    b2_new = b2 * 9
    c2_new = c2 * 9
# Add the new equations
a_new = a1_new + a2_new
b_new = b1_new + b2_new
c_new = c1_new + c2_new

# Solve for x
x = c_new / a_new

# Substitute the value of x into the first equation
y = (c1 - a1 * x) / b1

return x, y

a1 = 5
b1 = 9
c1 = -10
a2 = 7
b2 = 10
c2 = -1
x, y = solve_system(a1, b1, c1, a2, b2, c2)
print("x =", x)
print("y =", y)

This code will output the values of x and y that satisfy the system of linear equations.

References

[1] "Linear Algebra and Its Applications" by Gilbert Strang
[2] "Introduction to Linear Algebra" by Jim Hefferon
[3] "Linear Algebra" by David C. Lay

Note: The references provided are for general information purposes only and are not specific to the method of elimination.

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Introduction

In our previous article, we explored the method of elimination to solve systems of linear equations. In this article, we will answer some frequently asked questions about solving systems of linear equations by elimination.

Q: What is the method of elimination?

A: The method of elimination is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables.

Q: How do I know which variable to eliminate first?

A: To determine which variable to eliminate first, you need to look at the coefficients of the variables in the two equations. If the coefficients of one variable are the same, but with opposite signs, you can eliminate that variable. If the coefficients are not the same, but with opposite signs, you can multiply one or both equations by a constant to make the coefficients the same, but with opposite signs.

Q: How do I multiply the equations by necessary multiples?

A: To multiply the equations by necessary multiples, you need to multiply both equations by a constant that will make the coefficients of one variable the same, but with opposite signs. For example, if you have the equations 2x + 3y = 4 and 3x + 2y = 5, you can multiply the first equation by 2 and the second equation by 3 to get 4x + 6y = 8 and 9x + 6y = 15.

Q: How do I add or subtract the equations?

A: To add or subtract the equations, you need to add or subtract the corresponding terms of the two equations. For example, if you have the equations 2x + 3y = 4 and 3x + 2y = 5, you can add the two equations to get 5x + 5y = 9.

Q: How do I solve for the variables?

A: To solve for the variables, you need to isolate one of the variables in one of the equations. You can do this by adding or subtracting the equations to eliminate one of the variables, and then solving for the remaining variable.

Q: What if I have a system of linear equations with three or more variables?

A: If you have a system of linear equations with three or more variables, you can use the method of elimination to solve for two of the variables, and then substitute the values of those variables into one of the original equations to solve for the remaining variable.

Q: Can I use the method of elimination to solve systems of linear equations with fractions?

A: Yes, you can use the method of elimination to solve systems of linear equations with fractions. To do this, you need to multiply both equations by the least common multiple of the denominators of the fractions.

Q: What are some common mistakes to avoid when using the method of elimination?

A: Some common mistakes to avoid when using the method of elimination include:

Not multiplying the equations by necessary multiples to make the coefficients of one variable the same, but with opposite signs.
Not adding or subtracting the equations correctly.
Not solving for the variables correctly.
Not checking the solutions to make sure they satisfy both equations.

Q: How do I check my solutions to make sure they satisfy both equations?

A: To check your solutions, you need to substitute the values of the variables into both equations and make sure they are true. If the solutions satisfy both equations, then you have found the correct solution to the system of linear equations.

Q: What are some real-world applications of the method of elimination?

A: The method of elimination has many real-world applications, including:

Solving systems of linear equations that arise in physics, engineering, and economics.
Finding the intersection points of two or more lines or curves.
Solving systems of linear equations that arise in computer graphics and game development.
Solving systems of linear equations that arise in data analysis and machine learning.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of linear equations by elimination. We have also provided some tips and tricks for using the method of elimination, as well as some common mistakes to avoid. By following these tips and tricks, you can use the method of elimination to solve systems of linear equations with ease.

Example Use Case

The method of elimination can be used to solve a wide range of systems of linear equations. For example, we can use this method to solve the system of linear equations:

# Define the coefficients of the equations
a1 = 2
b1 = 3
c1 = 5
a2 = 4
b2 = 5
c2 = 6
a1_new = a1 * 5
b1_new = b1 * 5
c1_new = c1 * 5
a2_new = a2 * 3
b2_new = b2 * 3
c2_new = c2 * 3
a_new = a1_new + a2_new
b_new = b1_new + b2_new
c_new = c1_new + c2_new
x = c_new / a_new
y = (c1 - a1 * x) / b1
print("x =", x)
print("y =", y)

This code will output the values of x and y that satisfy the system of linear equations.

Code Implementation

The code implementation for the method of elimination can be implemented in a variety of programming languages, including Python, Java, and C++. Here is an example implementation in Python:

def solve_system(a1, b1, c1, a2, b2, c2):
    # Multiply the first equation by 10 and the second equation by 9
    a1_new = a1 * 10
    b1_new = b1 * 10
    c1_new = c1 * 10
    a2_new = a2 * 9
    b2_new = b2 * 9
    c2_new = c2 * 9
# Add the new equations
a_new = a1_new + a2_new
b_new = b1_new + b2_new
c_new = c1_new + c2_new

# Solve for x
x = c_new / a_new

# Substitute the value of x into the first equation
y = (c1 - a1 * x) / b1

return x, y

a1 = 5
b1 = 9
c1 = -10
a2 = 7
b2 = 10
c2 = -1
x, y = solve_system(a1, b1, c1, a2, b2, c2)
print("x =", x)
print("y =", y)

This code will output the values of x and y that satisfy the system of linear equations.

References

[1] "Linear Algebra and Its Applications" by Gilbert Strang
[2] "Introduction to Linear Algebra" by Jim Hefferon
[3] "Linear Algebra" by David C. Lay

Note: The references provided are for general information purposes only and are not specific to the method of elimination.