Solving Linear Equations Using the Substitution Method: xy14 and x-y4

In the realm of algebra, understanding how to solve pairs of linear equations is a fundamental skill. Let's explore a classic problem using the substitution method, where we are given the equations xy14 and x-y4.

Introduction to the Problem

We are given two equations:

First equation: xy 14 Second equation: x - y 4

The goal is to solve for the values of x and y using the substitution method.

Step-by-Step Solution

Step 1: Isolate one variable in one of the equations.

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

x - y 4

x y 4

Step 2: Substitute the expression for x into the first equation.

Now, we replace x in the first equation with the expression we found for x:

xy 14

(y 4)y 14

Step 3: Simplify the equation.

(y 4)y 14

y2 4y 14

y2 4y - 14 0

This is a quadratic equation in terms of y. We can solve it by factoring or using the quadratic formula.

Simplification by Substitution

Let's simplify the process instead by directly substituting the simpler form:

x - y 4

4y 14

2y 7

y 7/2

Step 4: Substitute the value of y back into one of the original equations.

Using the first equation:

xy 14

(7/2)y 14

y 14 / (7/2)

y 14 * (2/7)

y 4

Now, substitute y back into the simplified form of the second equation:

y 5

x - y 4

x - 5 4

x 9

Therefore, the solution to the system of equations is x 9 and y 5.

Verification

To verify the solution, we substitute the values x 9 and y 5 back into the original equations:

xy 14:

(9)(5) 45 ≠ 14 (Correction: 45 should be 14)

x - y 4:

9 - 5 4

Note: There is a discrepancy in the previous steps. Let's follow the correct steps:

From the second equation, x - y 4, we express x y 4. Substitute this into the first equation:

(y 4)y 14

y2 4y 14

y2 4y - 14 0

Solving this quadratic equation:

Using the quadratic formula, y (-b±√(b2-4ac))/2a

y (-4±√(42-4*1*14))/2*1

y (-4±√(16-56))/2

y (-4±√(-40))/2

This seems incorrect, let's solve directly:

2y 14 - 4

2y 10

y 5

Substitute y 5 into the second equation:

x - 5 4

x 9

Therefore, the correct solution is x 9 and y 5.

Conclusion

In conclusion, the method of substitution is a powerful tool in solving pairs of linear equations. By isolating one variable and substituting it into the other equation, we can easily find the values of the variables. The correct solution to the system of equations xy 14 and x - y 4 is x 9 and y 5.