Solving the System of Equations xy10 and x-y4 Using Algebraic Techniques

In this article, we will explore various methods to solve the system of equations xy10 and x-y4. This includes the use of substitution, elimination, and even solving quadratic equations. We will provide a detailed step-by-step guide for each method to ensure a clear understanding of the solution process.

Method 1: Substitution

Let's start by using the substitution method to solve the system of equations.

Step 1: Express one variable in terms of the other

From the second equation x-y4, we can express x in terms of y:

x y 4

Step 2: Substitute this expression into the first equation

Substitute x y 4 into the first equation xy10:

(y 4)y 10

Step 3: Solve the resulting equation

Expand and rearrange the equation:

y2 4y 10

y2 4y - 10 0

To solve this quadratic equation, we can use the quadratic formula:

y [-b ± √(b2 - 4ac)] / (2a)

where a 1, b 4, and c -10:

y [-4 ± √(16 40)] / 2

y [-4 ± √56] / 2

y [-4 ± 2√14] / 2

y -2 ± √14

Since y must be a real number, we discard the negative root:

y -2 √14

Step 4: Find the value of x

Substitute y -2 √14 into the expression x y 4:

x (-2 √14) 4

x 2 √14

Thus, the solution is:

x 2 √14, y -2 √14

Method 2: Elimination

Alternatively, we can use the elimination method to solve the system of equations.

Step 1: Add the two equations

Adding the equations xy10 and x-y4, we get:

2x 14

Step 2: Solve for x

x 7

Step 3: Substitute x7 into the second equation

x - y 4

7 - y 4

y 7 - 4

y 3

Thus, the solution is:

x 7, y 3

Method 3: Factoring

Another method to solve the system of equations involves factoring.

Step 1: Rearrange the equations

From the second equation, we express y in terms of x:

y x - 4

Step 2: Substitute this expression into the first equation

Substitute y x - 4 into the first equation:

x(x - 4) 10

x2 - 4x - 10 0

This is a quadratic equation. We can solve it using the quadratic formula:

x [4 ± √(16 40)] / 2

x [4 ± √56] / 2

x 2 ± √14

Step 3: Find the corresponding y values

Substitute x 2 √14 into the expression y x - 4:

y (2 √14) - 4

y -2 √14

Substitute x 2 - √14 into the expression y x - 4:

y (2 - √14) - 4

y -2 - √14

Thus, the solutions are:

x 2 √14, y -2 √14

x 2 - √14, y -2 - ?√14

Verification

To verify the solutions, substitute the values back into the original equations:

xy (2 √14)(-2 √14) 4 - (14) √28 10

x - y (2 √14) - (-2 √14) 2 2 4

Similarly, for the other set of solutions:

xy (2 - √14)(-2 - √14) 4 - (14) - √28 10

x - y (2 - √14) - (-2 - √14) 2 2 4

Both sets of solutions satisfy the original equations.