In the `xy` plane, line k has equation `y=2/9 x + 5`, and line n has equation `y=1/4 x +b`. If the lines intersect at the point with coordinates `(a, 2/3)` , what is the value of b ?

Answer

To find the value of \( b \) where the lines intersect at the point \( (a, \frac{2}{3}) \), we will follow these steps:

### Step 1: Substitute the intersection point into line \( k \)'s equation
The equation of line \( k \) is given by:
\[ y = \frac{2}{9}x + 5 \]
Since the lines intersect at the point \( (a, \frac{2}{3}) \), we can substitute \( y = \frac{2}{3} \) into line \( k \)'s equation:
\[ \frac{2}{3} = \frac{2}{9}a + 5 \]

### Step 2: Solve for \( a \)
To solve for \( a \), we first isolate \( a \):
1. Subtract 5 from both sides:
\[ \frac{2}{3} - 5 = \frac{2}{9}a \]
2. Convert 5 to a fraction with a denominator of 3:
\[ 5 = \frac{15}{3} \]
Thus,
\[ \frac{2}{3} - \frac{15}{3} = \frac{2}{9}a \]
This simplifies to:
\[ -\frac{13}{3} = \frac{2}{9}a \]
3. Multiply both sides by \( \frac{9}{2} \) to solve for \( a \):
\[ a = -\frac{13}{3} \cdot \frac{9}{2} = -\frac{117}{6} = -\frac{39}{2} \]

### Step 3: Substitute \( a \) into line \( n \)'s equation
The equation of line \( n \) is:
\[ y = \frac{1}{4}x + b \]
Substituting \( a = -\frac{39}{2} \) and \( y = \frac{2}{3} \):
\[ \frac{2}{3} = \frac{1}{4} \left(-\frac{39}{2}\right) + b \]

### Step 4: Solve for \( b \)
1. Calculate \( \frac{1}{4} \left(-\frac{39}{2}\right) \):
\[ \frac{1}{4} \cdot -\frac{39}{2} = -\frac{39}{8} \]
2. Substitute back into the equation:
\[ \frac{2}{3} = -\frac{39}{8} + b \]
3. Isolate \( b \):
\[ b = \frac{2}{3} + \frac{39}{8} \]
4. Convert \( \frac{2}{3} \) to a fraction with a denominator of 24:
\[ \frac{2}{3} = \frac{16}{24} \]
5. Convert \( \frac{39}{8} \) to a fraction with a denominator of 24:
\[ \frac{39}{8} = \frac{117}{24} \]
6. Now add the two fractions:
\[ b = \frac{16}{24} + \frac{117}{24} = \frac{133}{24} \]

### Final Answer
Thus, the value of \( b \) is:
\[ b = \frac{133}{24} \]