Solve graphically the following pairs of equations :
`2x+y=23`
`4x-y=19`

Answer

To solve the given pair of equations graphically, we will follow these steps:

### Step 1: Write down the equations
We have the following equations:
1. \( 2x + y = 23 \)
2. \( 4x - y = 19 \)

### Step 2: Convert the equations into slope-intercept form
We can rearrange both equations into the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

**For the first equation:**
\[ y = 23 - 2x \]

**For the second equation:**
\[ y = 4x - 19 \]

### Step 3: Find points for each equation
We will choose some values for \( x \) and calculate the corresponding \( y \) values for both equations.

**For \( 2x + y = 23 \):**
- Let \( x = 0 \):
\[ y = 23 - 2(0) = 23 \] → Point: \( (0, 23) \)
- Let \( x = 7 \):
\[ y = 23 - 2(7) = 9 \] → Point: \( (7, 9) \)
- Let \( x = 8 \):
\[ y = 23 - 2(8) = 7 \] → Point: \( (8, 7) \)

**For \( 4x - y = 19 \):**
- Let \( x = 5 \):
\[ y = 4(5) - 19 = 20 - 19 = 1 \] → Point: \( (5, 1) \)
- Let \( x = 4 \):
\[ y = 4(4) - 19 = 16 - 19 = -3 \] → Point: \( (4, -3) \)

### Step 4: Plot the points on a graph
Now we will plot the points we found on a graph.

**For the first equation \( 2x + y = 23 \):**
- Points: \( (0, 23) \), \( (7, 9) \), \( (8, 7) \)

**For the second equation \( 4x - y = 19 \):**
- Points: \( (5, 1) \), \( (4, -3) \)

### Step 5: Draw the lines
Using the points plotted, we will draw straight lines for both equations.

1. Draw a line through the points \( (0, 23) \), \( (7, 9) \), and \( (8, 7) \) for the first equation.
2. Draw a line through the points \( (5, 1) \) and \( (4, -3) \) for the second equation.

### Step 6: Identify the intersection point
The point where the two lines intersect is the solution to the system of equations.

From the graph, we can see that the lines intersect at the point \( (7, 9) \).

### Step 7: Write the solution
Thus, the solution to the system of equations is:
\[ x = 7, \quad y = 9 \]