If `x = a and y=b` is the solution of the pair of equation `x-y =5 and x+y =3,` find 'a' and 'b'.

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given the equations:

1. \( x - y = 5 \)
2. \( x + y = 3 \)

Since we know that \( x = a \) and \( y = b \), we can substitute these into the equations.

### Step 1: Substitute \( x \) and \( y \)

Substituting \( x = a \) and \( y = b \) into the first equation:

\[ a - b = 5 \tag{1} \]

Now substituting into the second equation:

\[ a + b = 3 \tag{2} \]

### Step 2: Solve the equations

Now we have a system of two equations:

1. \( a - b = 5 \)
2. \( a + b = 3 \)

We can solve these equations using the substitution or elimination method. Here, we will use the substitution method.

From equation (1), we can express \( a \) in terms of \( b \):

\[ a = b + 5 \tag{3} \]

### Step 3: Substitute \( a \) in equation (2)

Now we will substitute equation (3) into equation (2):

\[ (b + 5) + b = 3 \]

### Step 4: Simplify the equation

Combining like terms gives:

\[ 2b + 5 = 3 \]

### Step 5: Isolate \( b \)

To isolate \( b \), subtract 5 from both sides:

\[ 2b = 3 - 5 \]
\[ 2b = -2 \]

Now, divide both sides by 2:

\[ b = -1 \]

### Step 6: Find \( a \)

Now that we have \( b \), we can substitute it back into equation (3) to find \( a \):

\[ a = b + 5 \]
\[ a = -1 + 5 \]
\[ a = 4 \]

### Final Answer

Thus, the values of \( a \) and \( b \) are:

\[ a = 4, \quad b = -1 \]