Prediction: There Will Be One Solution(s) Because $\[ \begin{cases} y = 2x + 5 & \text{Equation 1} \\ 10 = Y + 3x & \text{Equation 2} \end{cases} \\]1. What Does \[$ Y \$\] Equal In Equation 1?2. Find The Value Of \[$ Y \$\].3.

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Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations, and we will use the given equations to predict the solution(s).

The Given Equations

The given system of linear equations is:

{y=2x+5Equation 110=y+3xEquation 2\begin{cases} y = 2x + 5 & \text{Equation 1} \\ 10 = y + 3x & \text{Equation 2} \end{cases}

Step 1: Understanding the Equations

To solve the system of linear equations, we need to understand the equations and identify the variables and constants. In Equation 1, the variable is yy, and the constant is 5. In Equation 2, the variable is also yy, and the constants are 10 and 3.

Step 2: Solving Equation 1 for yy

In Equation 1, we are given that y=2x+5y = 2x + 5. This equation is already solved for yy, so we can use it as is.

Step 3: Substituting Equation 1 into Equation 2

We can substitute the expression for yy from Equation 1 into Equation 2. This will give us an equation with only one variable, xx.

10=(2x+5)+3x10 = (2x + 5) + 3x

Step 4: Simplifying the Equation

We can simplify the equation by combining like terms.

10=5x+510 = 5x + 5

Step 5: Solving for xx

We can solve for xx by subtracting 5 from both sides of the equation.

5=5x5 = 5x

Step 6: Finding the Value of xx

We can find the value of xx by dividing both sides of the equation by 5.

x=1x = 1

Step 7: Finding the Value of yy

Now that we have the value of xx, we can substitute it into Equation 1 to find the value of yy.

y=2(1)+5y = 2(1) + 5

Step 8: Simplifying the Equation

We can simplify the equation by multiplying 2 and 1, and then adding 5.

y=2+5y = 2 + 5

Step 9: Finding the Value of yy

We can find the value of yy by adding 2 and 5.

y=7y = 7

Conclusion

In this article, we solved a system of two linear equations using the substitution method. We found that the value of xx is 1, and the value of yy is 7. This solution satisfies both equations in the system.

Discussion

The solution to the system of linear equations is unique, meaning that there is only one solution. This is because the two equations are linearly independent, meaning that they are not multiples of each other.

Final Answer

The final answer is 7\boxed{7}.

Related Topics

  • Solving systems of linear equations using the substitution method
  • Solving systems of linear equations using the elimination method
  • Graphing systems of linear equations
  • Solving systems of linear equations with three variables

Glossary

  • System of linear equations: A set of two or more linear equations that involve the same set of variables.
  • Linear equation: An equation in which the highest power of the variable(s) is 1.
  • Variable: A value that can change in a mathematical expression.
  • Constant: A value that does not change in a mathematical expression.
  • Substitution method: A method of solving systems of linear equations by substituting one equation into another.
  • Elimination method: A method of solving systems of linear equations by eliminating one variable by adding or subtracting the equations.
  • Graphing: A method of solving systems of linear equations by graphing the equations on a coordinate plane.

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Introduction

In our previous article, we solved a system of two linear equations using the substitution method. In this article, we will answer some frequently asked questions about solving systems of linear equations.

Q: What is a system of linear equations?

A system of linear equations is a set of two or more linear equations that involve the same set of variables.

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

A system of linear equations has a solution if the two equations are linearly independent, meaning that they are not multiples of each other.

Q: What is the difference between the substitution method and the elimination method?

The substitution method involves substituting one equation into another to solve for one variable, while the elimination method involves adding or subtracting the equations to eliminate one variable.

Q: Can a system of linear equations have more than one solution?

No, a system of linear equations can only have one solution, unless the two equations are identical, in which case there are infinitely many solutions.

Q: How do I know if a system of linear equations has infinitely many solutions?

A system of linear equations has infinitely many solutions if the two equations are identical.

Q: Can a system of linear equations have no solution?

Yes, a system of linear equations can have no solution if the two equations are inconsistent, meaning that they cannot be true at the same time.

Q: How do I know if a system of linear equations is inconsistent?

A system of linear equations is inconsistent if the two equations have no solution in common.

Q: What is the importance of solving systems of linear equations?

Solving systems of linear equations is important in many fields, including physics, engineering, economics, and computer science.

Q: How do I apply the concepts of solving systems of linear equations in real-life situations?

You can apply the concepts of solving systems of linear equations in real-life situations such as:

  • Finding the intersection of two lines
  • Determining the cost of producing a product
  • Calculating the interest rate on a loan
  • Solving optimization problems

Conclusion

Solving systems of linear equations is an important concept in mathematics and has many real-life applications. By understanding the concepts and methods of solving systems of linear equations, you can apply them to solve problems in various fields.

Final Answer

The final answer is 1\boxed{1}.

Related Topics

  • Solving systems of linear equations using the substitution method
  • Solving systems of linear equations using the elimination method
  • Graphing systems of linear equations
  • Solving systems of linear equations with three variables

Glossary

  • System of linear equations: A set of two or more linear equations that involve the same set of variables.
  • Linear equation: An equation in which the highest power of the variable(s) is 1.
  • Variable: A value that can change in a mathematical expression.
  • Constant: A value that does not change in a mathematical expression.
  • Substitution method: A method of solving systems of linear equations by substituting one equation into another.
  • Elimination method: A method of solving systems of linear equations by eliminating one variable by adding or subtracting the equations.
  • Graphing: A method of solving systems of linear equations by graphing the equations on a coordinate plane.