Solve The System Of Equations. 2 X − 4 Y = 18 X = − 1 \begin{array}{l} 2x - 4y = 18 \\ x = -1 \end{array} 2 X − 4 Y = 18 X = − 1 ​ Find The Values Of { X $}$ And { Y $}$.

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Introduction

In this article, we will explore how to solve a system of equations when one of the variables is already given. We will use the given value to find the other variable and solve for both x and y.

What is a System of Equations?

A system of equations is a set of two or more equations that contain the same variables. In this case, we have two equations with two variables, x and y.

The Given System of Equations

The given system of equations is:

2x−4y=18x=−1\begin{array}{l} 2x - 4y = 18 \\ x = -1 \end{array}

Substituting the Given Value

We are given the value of x, which is -1. We can substitute this value into the first equation to solve for y.

Step 1: Substitute x into the First Equation

Substituting x = -1 into the first equation, we get:

2(−1)−4y=182(-1) - 4y = 18

Step 2: Simplify the Equation

Simplifying the equation, we get:

−2−4y=18-2 - 4y = 18

Step 3: Add 2 to Both Sides

Adding 2 to both sides of the equation, we get:

−4y=20-4y = 20

Step 4: Divide Both Sides by -4

Dividing both sides of the equation by -4, we get:

y=−5y = -5

Conclusion

We have found the value of y, which is -5. Now that we have the value of y, we can substitute it back into the second equation to verify our solution.

Verifying the Solution

Substituting y = -5 into the second equation, we get:

x=−1x = -1

This confirms that our solution is correct.

Final Answer

The final answer is:

x=−1x = -1 y=−5y = -5

Discussion

In this article, we have shown how to solve a system of equations when one of the variables is already given. We used the given value to find the other variable and solve for both x and y. This is a useful technique to have in your mathematical toolkit, as it can be applied to a wide range of problems.

Example Use Cases

This technique can be applied to a wide range of problems, including:

  • Finding the intersection point of two lines
  • Solving a system of linear equations
  • Finding the solution to a quadratic equation

Tips and Tricks

When solving a system of equations, it's often helpful to:

  • Use substitution to solve for one variable
  • Use elimination to solve for the other variable
  • Check your solution by substituting it back into the original equations

Conclusion

In conclusion, solving a system of equations with a given value is a useful technique to have in your mathematical toolkit. By following the steps outlined in this article, you can solve for both x and y and find the solution to a wide range of problems.

Further Reading

If you're interested in learning more about solving systems of equations, I recommend checking out the following resources:

  • Khan Academy: Solving Systems of Equations
  • Mathway: Solving Systems of Equations
  • Wolfram Alpha: Solving Systems of Equations

References

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Introduction

In our previous article, we explored how to solve a system of equations when one of the variables is already given. We used the given value to find the other variable and solve for both x and y. In this article, we will answer some frequently asked questions about solving systems of equations with a given value.

Q&A

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that contain the same variables.

Q: How do I know which variable to substitute first?

A: You can substitute either variable first, but it's often easier to substitute the variable that is given.

Q: What if I get stuck on a step?

A: Don't worry! Take a step back and re-read the problem. You can also try drawing a diagram or using a different method to solve the problem.

Q: Can I use this method to solve a system of equations with more than two variables?

A: No, this method is only for systems of equations with two variables. If you have a system of equations with more than two variables, you will need to use a different method.

Q: How do I know if my solution is correct?

A: You can check your solution by substituting it back into the original equations. If the solution satisfies both equations, then it is correct.

Q: What if I get a negative value for one of the variables?

A: Don't worry! Negative values are perfectly valid in mathematics. Just make sure to include the negative sign when writing your final answer.

Q: Can I use this method to solve a system of equations with fractions?

A: Yes, you can use this method to solve a system of equations with fractions. Just make sure to simplify the fractions as you go along.

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

A: It depends on the problem. If you have a system of equations with two variables and one of the variables is given, then this method is a good choice. If you have a system of equations with more than two variables or no given values, then you will need to use a different method.

Example Problems

Problem 1

Solve the system of equations:

2x−4y=18x=3\begin{array}{l} 2x - 4y = 18 \\ x = 3 \end{array}

Solution

Substituting x = 3 into the first equation, we get:

2(3)−4y=182(3) - 4y = 18

Simplifying the equation, we get:

6−4y=186 - 4y = 18

Adding 4y to both sides of the equation, we get:

6=18+4y6 = 18 + 4y

Subtracting 18 from both sides of the equation, we get:

−12=4y-12 = 4y

Dividing both sides of the equation by 4, we get:

y=−3y = -3

Problem 2

Solve the system of equations:

x+2y=10x=−2\begin{array}{l} x + 2y = 10 \\ x = -2 \end{array}

Solution

Substituting x = -2 into the first equation, we get:

−2+2y=10-2 + 2y = 10

Simplifying the equation, we get:

2y=122y = 12

Dividing both sides of the equation by 2, we get:

y=6y = 6

Conclusion

In this article, we have answered some frequently asked questions about solving systems of equations with a given value. We have also provided example problems to help illustrate the method. Remember to always check your solution by substituting it back into the original equations.

Further Reading

If you're interested in learning more about solving systems of equations, I recommend checking out the following resources:

  • Khan Academy: Solving Systems of Equations
  • Mathway: Solving Systems of Equations
  • Wolfram Alpha: Solving Systems of Equations

References