9x-4y=2000. 7x-3y=2000. Solve By Substitution Method

by ADMIN 53 views

===========================================================

Introduction

In this article, we will explore the substitution method for solving a system of linear equations. The substitution method is a technique used to solve a system of linear equations by expressing one variable in terms of another and then substituting that expression into the other equation. This method is particularly useful when one of the equations can be easily solved for one of the variables.

The Problem

We are given two linear equations:

  • 9x - 4y = 2000
  • 7x - 3y = 2000

Our goal is to solve for the values of x and y that satisfy both equations.

Step 1: Solve One of the Equations for One Variable

Let's start by solving the first equation for x:

9x - 4y = 2000

We can add 4y to both sides of the equation to get:

9x = 2000 + 4y

Next, we can divide both sides of the equation by 9 to get:

x = (2000 + 4y) / 9

Step 2: Substitute the Expression into the Other Equation

Now that we have an expression for x in terms of y, we can substitute this expression into the second equation:

7x - 3y = 2000

Substituting x = (2000 + 4y) / 9 into this equation, we get:

7((2000 + 4y) / 9) - 3y = 2000

Step 3: Simplify the Equation

To simplify the equation, we can start by multiplying both sides of the equation by 9 to get rid of the fraction:

7(2000 + 4y) - 27y = 18000

Next, we can distribute the 7 to the terms inside the parentheses:

14000 + 28y - 27y = 18000

Step 4: Combine Like Terms

Now that we have simplified the equation, we can combine like terms:

14000 + y = 18000

Step 5: Solve for y

To solve for y, we can subtract 14000 from both sides of the equation:

y = 4000

Step 6: Substitute the Value of y into One of the Original Equations

Now that we have found the value of y, we can substitute this value into one of the original equations to find the value of x. Let's use the first equation:

9x - 4y = 2000

Substituting y = 4000 into this equation, we get:

9x - 16000 = 2000

Step 7: Solve for x

To solve for x, we can add 16000 to both sides of the equation:

9x = 16000 + 2000

9x = 18000

Next, we can divide both sides of the equation by 9 to get:

x = 2000

Conclusion

In this article, we used the substitution method to solve a system of linear equations. We started by solving one of the equations for one variable and then substituted that expression into the other equation. We then simplified the equation and solved for the value of y. Finally, we substituted the value of y into one of the original equations to find the value of x.

The Final Answer

The final answer is:

x = 2000 y = 4000

Example Use Case

The substitution method can be used to solve a wide range of systems of linear equations. For example, it can be used to solve systems of equations that arise in physics, engineering, and economics.

Tips and Variations

  • The substitution method can be used to solve systems of linear equations with any number of variables.
  • The substitution method can be used to solve systems of linear equations with any type of coefficients (e.g. integers, fractions, decimals).
  • The substitution method can be used to solve systems of linear equations with any type of constants (e.g. integers, fractions, decimals).

Further Reading

For more information on the substitution method, see the following resources: