Solving Simultaneous Equations: m2 - n2 13 and 2mn 84

When dealing with simultaneous equations, it's important to have a solid understanding of algebraic techniques, including factoring and the application of the quadratic formula. In this article, we will walk through the process of solving the given simultaneous equations:

Problem Statement

We are given the following system of equations:

m2 - n2 13 2mn 84

Step-by-Step Solution

Step 1: Rewrite the Equations

First, we will rewrite the equations to make them easier to handle.

Equation 1: (quad m^2 - n^2 13)

Equation 2: (quad 2mn 84)

Step 2: Factor and Simplify

Equation 1 can be factored using the difference of squares:

[quad (m n)(m - n) 13]

Equation 2 can be simplified to:

[quad mn 42]

Now, let's introduce new variables to help solve the system. Let (p mn) and (q m - n).

Step 3: Express (m) and (n) in Terms of (p) and (q)

From the definitions, we have:

[quad m frac{pq}{2} quad text{and} quad n frac{p - q}{2}]

Step 4: Substitute into the Equations

Substitute (p) and (q) into the equations:

[frac{pq}{2} cdot frac{p - q}{2} 42]

Simplify this to:

[frac{p^2 q - pq^2}{4} 42]

Then:

[frac{p^2 q}{4} - frac{pq^2}{4} 42]

Multiplying through by 4 to eliminate the fraction:

[quad p^2 q - pq^2 168quad text{(Equation 1)}

From the second equation, we have:

[quad q cdot p 13quad text{(Equation 2)}

Step 5: Solve the Equations

From Equation 2, we can express (q):

[quad q frac{13}{p}]

Substitute (q) into Equation 1:

[quad p^2 left( frac{13}{p} right) - p left( frac{13}{p} right)^2 168]

Simplify this to:

[quad 13p - frac{169}{p} 168]

Multiplying through by (p) to eliminate the fraction:

[quad 13p^2 - 169 168p]

Combining like terms:

[quad 13p^2 - 168p - 169 0]

Let (x p^2), so the equation becomes:

[quad 13x - 168x - 169 0]

[quad x^2 - 168x - 169 0]

Using the quadratic formula, (x frac{-b pm sqrt{b^2 - 4ac}}{2a}):

[quad x frac{168 pm sqrt{168^2 4 cdot 169}}{2}]

[quad x frac{168 pm sqrt{28224 676}}{2}]

[quad x frac{168 pm sqrt{28900}}{2}]

(sqrt{28900} 170), thus:

[quad x frac{168 pm 170}{2}]

This gives us two solutions:

[quad x_1 frac{338}{2} 169quad text{and} quad x_2 frac{-2}{2} -1 quad text{(not valid)}]

Therefore, (p^2 169), so (p 13) (since (p) must be positive).

Step 6: Find (q)

Substitute (p 13) back into Equation 2:

[quad q frac{13}{13} 1]

Step 7: Solve for (m) and (n)

Now we can find (m) and (n):

[quad mn 13 quad text{and} quad m - n 1]

Add these equations to find (2m):

[quad 2m 14 quad Rightarrow quad m 7]

Substitute (m 7) back to find (n):

[quad 7 - n 1 quad Rightarrow quad n 6]

Final Solution

The solution to the simultaneous equations is:

[quad m 7 quad text{and} quad n 6]

You can verify:

(7^2 - 6^2 49 - 36 13) (2 cdot 7 cdot 6 84)

Conclusion

Solving simultaneous equations involves a series of algebraic manipulations, including factoring, substitution, and the application of the quadratic formula. This step-by-step guide demonstrates how to systematically solve such equations and verify the solution.