The expressions ax + by has value 7 when x= 2 and y=1 . It has value 1 when x=-1 and y =1 . Find a and b.

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given the expressions and conditions provided.

### Step 1: Set up the equations based on the given conditions.

From the problem, we know:
1. When \( x = 2 \) and \( y = 1 \), the expression \( ax + by = 7 \).
- Substituting these values, we get:
\[ 2a + 1b = 7 \quad \text{(Equation 1)} \]

2. When \( x = -1 \) and \( y = 1 \), the expression \( ax + by = 1 \).
- Substituting these values, we get:
\[ -1a + 1b = 1 \quad \text{(Equation 2)} \]

### Step 2: Write down the equations.

We have the following two equations:
1. \( 2a + b = 7 \) (Equation 1)
2. \( -a + b = 1 \) (Equation 2)

### Step 3: Solve the equations simultaneously.

We can solve these two equations simultaneously. Let's eliminate \( b \) by subtracting Equation 2 from Equation 1.

Subtracting Equation 2 from Equation 1:
\[ (2a + b) - (-a + b) = 7 - 1 \]
This simplifies to:
\[ 2a + b + a - b = 6 \]
\[ 3a = 6 \]
Now, divide both sides by 3:
\[ a = 2 \]

### Step 4: Substitute \( a \) back into one of the equations to find \( b \).

Now that we have \( a = 2 \), we can substitute this value back into either Equation 1 or Equation 2 to find \( b \). Let's use Equation 1:
\[ 2(2) + b = 7 \]
This simplifies to:
\[ 4 + b = 7 \]
Subtracting 4 from both sides gives:
\[ b = 3 \]

### Final Answer:
Thus, the values of \( a \) and \( b \) are:
\[ a = 2, \quad b = 3 \]