Solve The System Of Equations:1. $y = -3x - 15$2. − 8 X + 7 Y = − 18 -8x + 7y = -18 − 8 X + 7 Y = − 18

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Introduction

In mathematics, 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. In this article, we will focus on solving a system of two linear equations using the substitution method and the elimination method. We will use the following system of equations as an example:

  1. y=−3x−15y = -3x - 15
  2. −8x+7y=−18-8x + 7y = -18

Understanding the System of Equations

Before we start solving the system of equations, let's understand what each equation represents. The first equation is a linear equation in two variables, xx and yy. It can be written in the slope-intercept form, y=mx+by = mx + b, where mm is the slope and bb is the y-intercept. In this case, the slope is −3-3 and the y-intercept is −15-15.

The second equation is also a linear equation in two variables, xx and yy. It can be written in the standard form, ax+by=cax + by = c, where aa, bb, and cc are constants. In this case, a=−8a = -8, b=7b = 7, and c=−18c = -18.

Substitution Method

The substitution method is a technique used to solve a system of linear equations by substituting one equation into the other. In this case, we can substitute the expression for yy from the first equation into the second equation.

Step 1: Substitute the expression for y into the second equation

Substitute y=−3x−15y = -3x - 15 into the second equation:

−8x+7(−3x−15)=−18-8x + 7(-3x - 15) = -18

Step 2: Simplify the equation

Expand and simplify the equation:

−8x−21x−105=−18-8x - 21x - 105 = -18

Combine like terms:

−29x−105=−18-29x - 105 = -18

Step 3: Solve for x

Add 105 to both sides of the equation:

−29x=87-29x = 87

Divide both sides of the equation by -29:

x=−3x = -3

Step 4: Find the value of y

Now that we have the value of xx, we can substitute it into one of the original equations to find the value of yy. We will use the first equation:

y=−3x−15y = -3x - 15

Substitute x=−3x = -3 into the equation:

y=−3(−3)−15y = -3(-3) - 15

Simplify the equation:

y=9−15y = 9 - 15

y=−6y = -6

Elimination Method

The elimination method is a technique used to solve a system of linear equations by eliminating one variable by adding or subtracting the equations. In this case, we can multiply the first equation by 7 and the second equation by 3 to eliminate the variable yy.

Step 1: Multiply the first equation by 7

Multiply the first equation by 7:

7y=−21x−1057y = -21x - 105

Step 2: Multiply the second equation by 3

Multiply the second equation by 3:

−24x+21y=−54-24x + 21y = -54

Step 3: Add the two equations

Add the two equations to eliminate the variable yy:

7y−24x+21y=−54−1057y - 24x + 21y = -54 - 105

Combine like terms:

28y−24x=−15928y - 24x = -159

Step 4: Solve for x

Now that we have the equation 28y−24x=−15928y - 24x = -159, we can solve for xx. However, we notice that the equation is not in the standard form. We can rewrite the equation as:

−24x=−159−28y-24x = -159 - 28y

Divide both sides of the equation by -24:

x=−159−28y−24x = \frac{-159 - 28y}{-24}

Simplify the equation:

x=159+28y24x = \frac{159 + 28y}{24}

Step 5: Find the value of y

Now that we have the value of xx, we can substitute it into one of the original equations to find the value of yy. We will use the first equation:

y=−3x−15y = -3x - 15

Substitute x=159+28y24x = \frac{159 + 28y}{24} into the equation:

y=−3(159+28y24)−15y = -3\left(\frac{159 + 28y}{24}\right) - 15

Simplify the equation:

y=−477−84y24−15y = \frac{-477 - 84y}{24} - 15

y=−477−84y−36024y = \frac{-477 - 84y - 360}{24}

y=−837−84y24y = \frac{-837 - 84y}{24}

24y=−837−84y24y = -837 - 84y

108y=−837108y = -837

y=−837108y = -\frac{837}{108}

y=−7.75y = -7.75

Conclusion

In this article, we solved a system of two linear equations using the substitution method and the elimination method. We found that the value of xx is −3-3 and the value of yy is −6-6 using the substitution method, and the value of xx is 159+28y24\frac{159 + 28y}{24} and the value of yy is −7.75-7.75 using the elimination method.

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Q: What is a system of linear equations?

A: 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 are the two main methods for solving systems of linear equations?

A: The two main methods for solving systems of linear equations are the substitution method and the elimination method.

Q: What is the substitution method?

A: The substitution method is a technique used to solve a system of linear equations by substituting one equation into the other.

Q: What is the elimination method?

A: The elimination method is a technique used to solve a system of linear equations by eliminating one variable by adding or subtracting the equations.

Q: How do I choose between the substitution method and the elimination method?

A: You can choose between the substitution method and the elimination method based on the form of the equations and the variables involved. If the equations are in the form of y=mx+by = mx + b, it is easier to use the substitution method. If the equations are in the form of ax+by=cax + by = c, it is easier to use the elimination method.

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

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

  • Not checking the solution to make sure it satisfies both equations
  • Not using the correct method for the form of the equations
  • Not simplifying the equations before solving
  • Not checking for extraneous solutions

Q: How do I check if a solution is extraneous?

A: To check if a solution is extraneous, you can substitute the solution into both equations and check if it satisfies both equations. If it does not satisfy both equations, it is an extraneous solution.

Q: Can I use other methods to solve systems of linear equations?

A: Yes, there are other methods to solve systems of linear equations, such as the graphing method and the matrix method. However, these methods are not as commonly used as the substitution method and the elimination method.

Q: How do I graph a system of linear equations?

A: To graph a system of linear equations, you can plot the equations on a coordinate plane and find the point of intersection. The point of intersection is the solution to the system.

Q: What is the matrix method?

A: The matrix method is a method used to solve systems of linear equations by representing the equations as matrices and using matrix operations to find the solution.

Q: Is the matrix method more efficient than the substitution method and the elimination method?

A: The matrix method can be more efficient than the substitution method and the elimination method for large systems of linear equations. However, it requires a good understanding of matrix operations and can be more difficult to use for small systems of linear equations.

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

A: Yes, you can use technology such as graphing calculators and computer software to solve systems of linear equations. These tools can help you to visualize the equations and find the solution more easily.

Q: How do I choose the right technology for solving systems of linear equations?

A: You can choose the right technology for solving systems of linear equations based on your needs and the type of equations you are working with. Some popular options include graphing calculators, computer software, and online tools.

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

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

  • Physics and engineering: Solving systems of linear equations is used to model real-world problems such as motion, forces, and energies.
  • Economics: Solving systems of linear equations is used to model economic systems and make predictions about the behavior of markets.
  • Computer science: Solving systems of linear equations is used in computer graphics, game development, and other areas of computer science.
  • Data analysis: Solving systems of linear equations is used in data analysis to model relationships between variables and make predictions about future trends.

Q: Can I use solving systems of linear equations to solve other types of problems?

A: Yes, solving systems of linear equations can be used to solve other types of problems, such as:

  • Quadratic equations: Solving systems of linear equations can be used to solve quadratic equations by representing them as a system of linear equations.
  • Polynomial equations: Solving systems of linear equations can be used to solve polynomial equations by representing them as a system of linear equations.
  • Differential equations: Solving systems of linear equations can be used to solve differential equations by representing them as a system of linear equations.

Q: How do I practice solving systems of linear equations?

A: You can practice solving systems of linear equations by:

  • Working on practice problems and exercises
  • Using online resources and tools to practice solving systems of linear equations
  • Joining a study group or working with a tutor to practice solving systems of linear equations
  • Using real-world examples and applications to practice solving systems of linear equations.