Solve The System Of Equations:1. { -3x - 3y = 3$}$2. { Y = -5x - 17$}$

Introduction

In mathematics, a system of equations is a set of two or more 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 with two variables. We will use the given equations:

1. ${-3x - 3y = 3}$
2. ${y = -5x - 17}$

Understanding the Equations

The first equation is a linear equation in two variables, x and y. It can be written in the standard form as:

${ax + by = c}$

where a, b, and c are constants. In this case, a = -3, b = -3, and c = 3.

The second equation is also a linear equation in two variables, x and y. It can be written in the standard form as:

${y = mx + b}$

where m is the slope of the line and b is the y-intercept. In this case, m = -5 and b = -17.

Substitution Method

One of the methods to solve a system of equations is the substitution method. This method involves substituting the expression for one variable from one equation into the other equation. In this case, we can substitute the expression for y from the second equation into the first equation.

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

Substitute the expression for y from the second equation into the first equation:

${-3x - 3y = 3}$

${y = -5x - 17}$

${-3x - 3(-5x - 17) = 3}$

Step 3: Simplify the equation

Simplify the equation by combining like terms:

${-3x + 15x + 51 = 3}$

${12x + 51 = 3}$

Step 4: Isolate the variable x

Subtract 51 from both sides of the equation:

${12x = -48}$

Divide both sides of the equation by 12:

${x = -4}$

Step 5: Find the value of y

Now that we have the value of x, substitute it into one of the original equations to find the value of y. We will use the second equation:

${y = -5x - 17}$

${y = -5(-4) - 17}$

${y = 20 - 17}$

${y = 3}$

Conclusion

In this article, we solved a system of two linear equations with two variables using the substitution method. We substituted the expression for y from the second equation into the first equation, simplified the equation, isolated the variable x, and found the value of y. The final solution is x = -4 and y = 3.

Example Use Cases

Solving systems of equations has many practical applications in real-life situations. Here are a few examples:

• Physics and Engineering: Systems of equations are used to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
• Computer Science: Systems of equations are used in computer graphics, game development, and machine learning.
• Economics: Systems of equations are used to model economic systems, such as supply and demand, and to make predictions about future economic trends.

Tips and Tricks

Here are a few tips and tricks to help you solve systems of equations:

• Use the substitution method: The substitution method is a powerful tool for solving systems of equations. It involves substituting the expression for one variable from one equation into the other equation.
• Simplify the equation: Simplifying the equation by combining like terms can make it easier to isolate the variable.
• Check your work: Always check your work by plugging the values of x and y back into the original equations to make sure they are true.

Conclusion

Frequently Asked Questions

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

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

A: There are several methods to solve a system of equations, including the substitution method, the elimination method, and the graphing method. The choice of method depends on the type of equations and the variables involved.

Q: What is the substitution method?

A: The substitution method is a method of solving a system of equations by substituting the expression for one variable from one equation into the other equation.

Q: How do I use the substitution method?

A: To use the substitution method, follow these steps:

1. Identify the equations and the variables involved.
2. Choose one of the equations and solve it for one of the variables.
3. Substitute the expression for the variable into the other equation.
4. Simplify the equation and solve for the other variable.
5. Check your work by plugging the values of the variables back into the original equations.

Q: What is the elimination method?

A: The elimination method is a method of solving a system of equations by adding or subtracting the equations to eliminate one of the variables.

Q: How do I use the elimination method?

A: To use the elimination method, follow these steps:

1. Identify the equations and the variables involved.
2. Choose two of the equations and multiply them by necessary multiples such that the coefficients of one of the variables are the same.
3. Add or subtract the equations to eliminate one of the variables.
4. Simplify the equation and solve for the other variable.
5. Check your work by plugging the values of the variables back into the original equations.

Q: What is the graphing method?

A: The graphing method is a method of solving a system of equations by graphing the equations on a coordinate plane and finding the point of intersection.

Q: How do I use the graphing method?

A: To use the graphing method, follow these steps:

1. Identify the equations and the variables involved.
2. Graph the equations on a coordinate plane.
3. Find the point of intersection of the two graphs.
4. Check your work by plugging the values of the variables back into the original equations.

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

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

• Not checking your work: Always check your work by plugging the values of the variables back into the original equations.
• Not using the correct method: Choose the correct method for the type of equations and the variables involved.
• Not simplifying the equation: Simplify the equation by combining like terms to make it easier to isolate the variable.
• Not being careful with signs: Be careful with signs when adding or subtracting equations.

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

A: A system of equations has a solution if the equations are consistent and the variables are related in a way that allows for a unique solution.

Q: What is the difference between a system of equations and a system of inequalities?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. A system of inequalities is a set of two or more inequalities that are solved simultaneously to find the values of the variables.

Conclusion

Solving systems of equations is an important skill in mathematics and has many practical applications in real-life situations. By understanding the different methods of solving systems of equations, including the substitution method, the elimination method, and the graphing method, you can solve systems of equations with ease. Remember to always check your work and to use the correct method for the type of equations and the variables involved.