Solve graphically the following pairs of equations :
`(x+1)/4=2/3(1-2y)`
`(2+5y)/3=x/7-2`

Answer

To solve the given pair of equations graphically, we will first rewrite the equations in a more manageable form, derive the equations of the lines, and then plot them on a Cartesian plane to find their intersection point.

### Step 1: Rewrite the first equation
The first equation is given as:
\[ \frac{x+1}{4} = \frac{2}{3}(1 - 2y) \]
To eliminate the fractions, we can cross-multiply:
\[ 3(x + 1) = 4 \cdot 2(1 - 2y) \]
This simplifies to:
\[ 3x + 3 = 8(1 - 2y) \]
Expanding the right side:
\[ 3x + 3 = 8 - 16y \]
Rearranging gives:
\[ 16y = 8 - 3x - 3 \]
\[ 16y = 5 - 3x \]
Dividing by 16 to express \(y\) in terms of \(x\):
\[ y = \frac{5 - 3x}{16} \]

### Step 2: Rewrite the second equation
The second equation is given as:
\[ \frac{2 + 5y}{3} = \frac{x}{7} - 2 \]
We can first simplify the right side:
\[ \frac{2 + 5y}{3} = \frac{x - 14}{7} \]
Cross-multiplying gives:
\[ 7(2 + 5y) = 3(x - 14) \]
Expanding both sides:
\[ 14 + 35y = 3x - 42 \]
Rearranging gives:
\[ 35y = 3x - 42 - 14 \]
\[ 35y = 3x - 56 \]
Dividing by 35 to express \(y\) in terms of \(x\):
\[ y = \frac{3x - 56}{35} \]

### Step 3: Find points to plot
Now we have two equations:
1. \(y = \frac{5 - 3x}{16}\)
2. \(y = \frac{3x - 56}{35}\)

We can choose several values for \(x\) to find corresponding \(y\) values.

**For the first equation:**
- If \(x = 1\):
\[ y = \frac{5 - 3(1)}{16} = \frac{2}{16} = \frac{1}{8} \]
- If \(x = 7\):
\[ y = \frac{5 - 3(7)}{16} = \frac{5 - 21}{16} = \frac{-16}{16} = -1 \]

**For the second equation:**
- If \(x = 1\):
\[ y = \frac{3(1) - 56}{35} = \frac{3 - 56}{35} = \frac{-53}{35} \]
- If \(x = 7\):
\[ y = \frac{3(7) - 56}{35} = \frac{21 - 56}{35} = \frac{-35}{35} = -1 \]

### Step 4: Plotting the points
We now have the following points to plot:
1. From the first equation: \((1, \frac{1}{8})\) and \((7, -1)\)
2. From the second equation: \((1, \frac{-53}{35})\) and \((7, -1)\)

### Step 5: Draw the lines and find the intersection
- Plot the points on a Cartesian plane.
- Draw the line for the first equation using the points \((1, \frac{1}{8})\) and \((7, -1)\).
- Draw the line for the second equation using the points \((1, \frac{-53}{35})\) and \((7, -1)\).
- The intersection point of the two lines is the solution to the system of equations.

### Final Answer
The intersection point is \((7, -1)\).

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