Solve Each System Of Equations Algebraically.15) { Y = − ( X − 2 ) 2 + 5 Y = − X + 1 \begin{cases} y = -(x-2)^2 + 5 \\ y = -x + 1 \end{cases} { Y = − ( X − 2 ) 2 + 5 Y = − X + 1 ​

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Introduction

Solving systems of equations algebraically is a fundamental concept in mathematics that involves finding the solution to a set of equations where the variables are related to each other. In this article, we will focus on solving a system of equations using algebraic methods. We will use the given system of equations as an example to demonstrate the steps involved in solving it.

The System of Equations

The given system of equations is:

{y=−(x−2)2+5y=−x+1\begin{cases} y = -(x-2)^2 + 5 \\ y = -x + 1 \end{cases}

Step 1: Write Down the Equations

The first step in solving a system of equations is to write down the equations. In this case, we have two equations:

  1. y=−(x−2)2+5y = -(x-2)^2 + 5
  2. y=−x+1y = -x + 1

Step 2: Set the Equations Equal to Each Other

To solve the system of equations, we need to set the two equations equal to each other. This is because both equations are equal to yy, so we can set them equal to each other.

−(x−2)2+5=−x+1-(x-2)^2 + 5 = -x + 1

Step 3: Expand and Simplify the Equation

The next step is to expand and simplify the equation. We can start by expanding the squared term:

−(x2−4x+4)+5=−x+1-(x^2 - 4x + 4) + 5 = -x + 1

Step 4: Combine Like Terms

Next, we can combine like terms:

−x2+4x−4+5=−x+1-x^2 + 4x - 4 + 5 = -x + 1

Step 5: Simplify the Equation

Now, we can simplify the equation by combining the constants:

−x2+4x+1=−x+1-x^2 + 4x + 1 = -x + 1

Step 6: Move All Terms to One Side

To solve for xx, we need to move all terms to one side of the equation. We can do this by subtracting −x-x from both sides:

−x2+4x+1+x=1-x^2 + 4x + 1 + x = 1

Step 7: Simplify the Equation

Now, we can simplify the equation by combining like terms:

−x2+5x+1=1-x^2 + 5x + 1 = 1

Step 8: Subtract 1 from Both Sides

Next, we can subtract 1 from both sides to get:

−x2+5x=0-x^2 + 5x = 0

Step 9: Factor Out the Common Term

Now, we can factor out the common term −x-x:

−x(x−5)=0-x(x - 5) = 0

Step 10: Solve for x

Finally, we can solve for xx by setting each factor equal to 0:

−x=0-x = 0 or x−5=0x - 5 = 0

Step 11: Solve for x

Solving for xx, we get:

x=0x = 0 or x=5x = 5

Step 12: Find the Corresponding y-Values

Now that we have found the values of xx, we can find the corresponding yy-values by substituting the values of xx into one of the original equations. We will use the first equation:

y=−(x−2)2+5y = -(x-2)^2 + 5

Step 13: Substitute x = 0 into the Equation

Substituting x=0x = 0 into the equation, we get:

y=−(0−2)2+5y = -(0-2)^2 + 5

Step 14: Simplify the Equation

Simplifying the equation, we get:

y=−4+5y = -4 + 5

Step 15: Solve for y

Solving for yy, we get:

y=1y = 1

Step 16: Substitute x = 5 into the Equation

Substituting x=5x = 5 into the equation, we get:

y=−(5−2)2+5y = -(5-2)^2 + 5

Step 17: Simplify the Equation

Simplifying the equation, we get:

y=−9+5y = -9 + 5

Step 18: Solve for y

Solving for yy, we get:

y=−4y = -4

Conclusion

In this article, we have solved a system of equations algebraically using the given system of equations as an example. We have followed the steps involved in solving a system of equations, including writing down the equations, setting the equations equal to each other, expanding and simplifying the equation, combining like terms, simplifying the equation, moving all terms to one side, factoring out the common term, solving for xx, finding the corresponding yy-values, and substituting the values of xx into one of the original equations. We have found the values of xx and the corresponding yy-values, which are x=0,y=1x = 0, y = 1 and x=5,y=−4x = 5, y = -4.

Final Answer

Introduction

Solving systems of equations algebraically is a fundamental concept in mathematics that involves finding the solution to a set of equations where the variables are related to each other. In this article, we will provide a Q&A guide to help you understand the steps involved in solving a system of equations algebraically.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve the same variables. In other words, it is a set of equations that are related to each other.

Q: How do I solve a system of equations algebraically?

A: To solve a system of equations algebraically, you need to follow these steps:

  1. Write down the equations.
  2. Set the equations equal to each other.
  3. Expand and simplify the equation.
  4. Combine like terms.
  5. Simplify the equation.
  6. Move all terms to one side.
  7. Factor out the common term.
  8. Solve for x.
  9. Find the corresponding y-values.
  10. Substitute the values of x into one of the original equations.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3 = 0 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, or by multiplying or dividing both sides of the equation by the same value.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. This formula will give you two possible values for x.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. It is given by: x = (-b ± √(b^2 - 4ac)) / 2a.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. You will then get two possible values for x.

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

A: A system of linear equations is a set of two or more linear equations that involve the same variables. A system of quadratic equations, on the other hand, is a set of two or more quadratic equations that involve the same variables.

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

A: To solve a system of linear equations, you need to follow the steps involved in solving a system of equations algebraically.

Q: How do I solve a system of quadratic equations?

A: To solve a system of quadratic equations, you need to use the quadratic formula to solve each equation separately, and then find the intersection of the two equations.

Conclusion

In this article, we have provided a Q&A guide to help you understand the steps involved in solving a system of equations algebraically. We have also discussed the difference between a linear equation and a quadratic equation, and how to solve each type of equation. We hope that this guide has been helpful in understanding the concept of solving systems of equations algebraically.

Final Answer

The final answer is \boxed{No final answer, as this is a Q&A guide.}