Solve The System Of Equations:${ \begin{array}{l} 3x - 2y = 0 \ x + Y = -5 \end{array} }$

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Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It 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. We will use the given system of equations as an example and provide a step-by-step guide on how to solve it.

The System of Equations

The given system of equations is:

3x−2y=0x+y=−5\begin{array}{l} 3x - 2y = 0 \\ x + y = -5 \end{array}

Understanding the Problem

To solve this system of equations, we need to find the values of x and y that satisfy both equations simultaneously. In other words, we need to find the point of intersection of the two lines represented by the equations.

Method 1: Substitution Method

One way to solve this system of equations is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Step 1: Solve the Second Equation for y

We can solve the second equation for y by isolating y on one side of the equation.

x+y=−5x + y = -5

y=−5−xy = -5 - x

Step 2: Substitute the Expression for y into the First Equation

Now that we have an expression for y, we can substitute it into the first equation.

3x−2y=03x - 2y = 0

3x−2(−5−x)=03x - 2(-5 - x) = 0

Step 3: Simplify the Equation

Simplifying the equation, we get:

3x+10+2x=03x + 10 + 2x = 0

5x+10=05x + 10 = 0

Step 4: Solve for x

Now, we can solve for x by isolating x on one side of the equation.

5x=−105x = -10

x=−2x = -2

Step 5: Find the Value of y

Now that we have the value of x, we can find the value of y by substituting x into the expression for y.

y=−5−xy = -5 - x

y=−5−(−2)y = -5 - (-2)

y=−3y = -3

Method 2: Elimination Method

Another way to solve this system of equations is by using the elimination method. This method involves eliminating one variable by adding or subtracting the equations.

Step 1: Multiply the Two Equations by Necessary Multiples

To eliminate one variable, we need to multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same.

3x−2y=03x - 2y = 0

x+y=−5x + y = -5

Multiplying the second equation by 2, we get:

2x+2y=−102x + 2y = -10

Step 2: Add the Two Equations

Now that we have the coefficients of y's in both equations the same, we can add the two equations to eliminate y.

(3x−2y)+(2x+2y)=0+(−10)(3x - 2y) + (2x + 2y) = 0 + (-10)

5x=−105x = -10

Step 3: Solve for x

Now, we can solve for x by isolating x on one side of the equation.

x=−2x = -2

Step 4: Find the Value of y

Now that we have the value of x, we can find the value of y by substituting x into one of the original equations.

x+y=−5x + y = -5

−2+y=−5-2 + y = -5

y=−3y = -3

Conclusion

In this article, we have solved a system of two linear equations with two variables using two different methods: substitution method and elimination method. We have found the values of x and y that satisfy both equations simultaneously, which is the point of intersection of the two lines represented by the equations. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation, while the elimination method involves eliminating one variable by adding or subtracting the equations.

Applications of Solving Systems of Linear Equations

Solving systems of linear equations has numerous applications in various fields, including:

  • Physics and Engineering: Solving systems of linear equations is essential in physics and engineering to model real-world problems, such as motion, forces, and energies.
  • Computer Science: Solving systems of linear equations is used in computer science to solve problems in computer graphics, game development, and machine learning.
  • Economics: Solving systems of linear equations is used in economics to model economic systems, such as supply and demand, and to make predictions about economic trends.
  • Biology: Solving systems of linear equations is used in biology to model population dynamics, such as the growth and decline of populations.

Tips and Tricks

Here are some tips and tricks to help you solve systems of linear equations:

  • Use the substitution method when one equation is easy to solve for one variable.
  • Use the elimination method when the coefficients of one variable are the same in both equations.
  • Check your work by plugging the values of x and y back into the original equations.
  • Use a graphing calculator or computer software to visualize the system of equations and find the point of intersection.

Conclusion

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is a linear equation, which means it is an equation in which the highest power of the variable(s) is 1.

Q: How do I know if a system of linear equations has a solution?

A: A system of linear equations has a solution if and only if the lines represented by the equations intersect at a