Solve The System Of Equations:1. { Y - 4x = 8 $}$2. { \frac{y + 32}{3x - 6} = 4 $}$

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Introduction

Solving a system of equations involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables. The first equation is in the form of y - 4x = 8, and the second equation is in the form of (y + 32)/(3x - 6) = 4. We will use algebraic methods to solve this system of equations.

Step 1: Write Down the Given Equations

The given equations are:

  1. y - 4x = 8
  2. (y + 32)/(3x - 6) = 4

Step 2: Simplify the Second Equation

To simplify the second equation, we can multiply both sides by (3x - 6) to eliminate the fraction.

(y + 32) = 4(3x - 6)

Expanding the right-hand side, we get:

y + 32 = 12x - 24

Subtracting 32 from both sides, we get:

y = 12x - 56

Step 3: Equate the Two Expressions for y

Now that we have simplified the second equation, we can equate the two expressions for y.

y - 4x = 8 ... (Equation 1) y = 12x - 56 ... (Equation 2)

Substituting Equation 2 into Equation 1, we get:

12x - 56 - 4x = 8

Combine like terms:

8x - 56 = 8

Adding 56 to both sides:

8x = 64

Dividing both sides by 8:

x = 8

Step 4: Find the Value of y

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. We will use Equation 2:

y = 12x - 56 y = 12(8) - 56 y = 96 - 56 y = 40

Step 5: Check the Solution

To check our solution, we can substitute the values of x and y into both original equations.

Equation 1: y - 4x = 8 40 - 4(8) = 8 40 - 32 = 8 8 = 8 (True)

Equation 2: (y + 32)/(3x - 6) = 4 (40 + 32)/(3(8) - 6) = 4 72/18 = 4 4 = 4 (True)

Since both equations are true, our solution is correct.

Conclusion

In this article, we solved a system of two linear equations with two variables using algebraic methods. We simplified the second equation, equated the two expressions for y, and found the values of x and y. We then checked our solution by substituting the values of x and y into both original equations. The solution was found to be correct.

Tips and Variations

  • To solve a system of equations with more than two variables, you can use methods such as substitution or elimination.
  • To solve a system of equations with non-linear equations, you may need to use numerical methods or graphing techniques.
  • To solve a system of equations with multiple solutions, you may need to use methods such as finding the intersection of multiple graphs.

Common Mistakes

  • Failing to simplify the second equation before equating the two expressions for y.
  • Failing to check the solution by substituting the values of x and y into both original equations.
  • Using the wrong method to solve the system of equations.

Real-World Applications

  • Solving systems of equations is a common problem in physics, engineering, and economics.
  • In physics, systems of equations are used to model the motion of objects and the behavior of electrical circuits.
  • In engineering, systems of equations are used to design and optimize systems such as bridges and buildings.
  • In economics, systems of equations are used to model the behavior of markets and the impact of policy changes.

Glossary

  • System of equations: A set of two or more equations that involve two or more variables.
  • Linear equation: An equation in which the highest power of the variable is 1.
  • Non-linear equation: An equation in which the highest power of the variable is greater than 1.
  • Substitution method: A method of solving a system of equations by substituting one equation into another.
  • Elimination method: A method of solving a system of equations by eliminating one variable by adding or subtracting the equations.
  • Numerical method: A method of solving a system of equations using numerical techniques such as the Newton-Raphson method.
  • Graphing technique: A method of solving a system of equations by graphing the equations on a coordinate plane.
    Solve the System of Equations: Q&A =====================================

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve two or more variables. In this article, we solved a system of two linear equations with two variables.

Q: What are the different methods to solve a system of equations?

A: There are several methods to solve a system of equations, including:

  • Substitution method: A method of solving a system of equations by substituting one equation into another.
  • Elimination method: A method of solving a system of equations by eliminating one variable by adding or subtracting the equations.
  • Numerical method: A method of solving a system of equations using numerical techniques such as the Newton-Raphson method.
  • Graphing technique: A method of solving a system of equations by graphing the equations on a coordinate plane.

Q: What is the difference between a linear equation and a non-linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3y = 5 is a linear equation. A non-linear equation is an equation in which the highest power of the variable is greater than 1. For example, x^2 + 3y = 5 is a non-linear equation.

Q: How do I know which method to use to solve a system of equations?

A: The choice of method depends on the type of equations and the number of variables. If the equations are linear and there are two variables, the substitution or elimination method may be the best choice. If the equations are non-linear or there are more than two variables, a numerical method or graphing technique may be more suitable.

Q: What are some common mistakes to avoid when solving a system of equations?

A: Some common mistakes to avoid when solving a system of equations include:

  • Failing to simplify the second equation before equating the two expressions for y.
  • Failing to check the solution by substituting the values of x and y into both original equations.
  • Using the wrong method to solve the system of equations.

Q: How do I check my solution to a system of equations?

A: To check your solution to a system of equations, substitute the values of x and y into both original equations. If the equations are true, then your solution is correct.

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

A: Solving systems of equations has many real-world applications, including:

  • Physics: Solving systems of equations is used to model the motion of objects and the behavior of electrical circuits.
  • Engineering: Solving systems of equations is used to design and optimize systems such as bridges and buildings.
  • Economics: Solving systems of equations is used to model the behavior of markets and the impact of policy changes.

Q: Can I use a calculator to solve a system of equations?

A: Yes, you can use a calculator to solve a system of equations. Many calculators have built-in functions for solving systems of equations, such as the substitution and elimination methods.

Q: How do I graph a system of equations?

A: To graph a system of equations, plot the two equations on a coordinate plane. The point of intersection of the two graphs is the solution to the system of equations.

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

A: Some tips for solving systems of equations include:

  • Read the problem carefully: Make sure you understand what the problem is asking for.
  • Simplify the equations: Simplify the equations before solving the system.
  • Use the correct method: Choose the correct method for solving the system of equations.
  • Check your solution: Check your solution by substituting the values of x and y into both original equations.