Question

Use the substitution method to solve the system.

{m+n=5m−n=3

What is the solution?

There is no solution.

The solution is (4, 1).

There are an infinite number of solutions.

The solution is (2, −1).

Answer 1

m + n = 5......m = 5 - n
m - n = 3

5 - n - n = 3
5 - 2n = 3
5 - 3 = 2n
2 = 2n
2/2 = n
1 = n

m - n = 3
m - 1 = 3
m = 3 + 1
m = 4

solution is (4,1)

Answer 2

The system of the equation has a unique solution and the solution of the equation is (4, 1).

Given equation

• equation 1, m+n=5
• equation 2, 5m−n=3

From equation 1,

solving for m,

$m+n=5\\m = 5-n$

From equation 2,

$m-n=3$

substituting the value of m,

$m-n=3\\(5-n)-n=3\\5-n-n=3\\-2n=3-5\\-2n=-2\\\\n=\dfrac{-2}{-2}\\\\n=1$

Substituting the value of n in all the equation 1,

$m+n=5\\m+(1)=5\\m=5-1\\m=4$

As the value of both the variables is unique also neither of the equations is in the ratio, therefore, the system has a unique solution.

Verification

To verify that the system of equations has a unique solution we can draw a graph.

Hence, the system of the equation has a unique solution and the solution of the equation is (4, 1).

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