Question

Solve graphically:
\[ 2x + 3y = 5 \] \[ 5x - 4y + 22 = 0 \]

Hint:

In graphical method, the solution of a pair of linear equations is the point where the two lines intersect.

Answer 1

Step 1: Write both equations clearly.}
The given pair of linear equations is:
\[ 2x + 3y = 5 \] \[ 5x - 4y + 22 = 0 \] The second equation can be written as:
\[ 5x - 4y = -22 \]
Step 2: Understand graphical solution.}
To solve graphically, we draw the lines represented by both equations on the same graph. The coordinates of their point of intersection give the required solution.

Step 3: Find points for the first line.}
For the equation \(2x + 3y = 5\):
If \(x = 1\), then
\[ 2(1) + 3y = 5 \] \[ 2 + 3y = 5 \] \[ 3y = 3 \] \[ y = 1 \] So one point is \((1,1)\).
If \(x = 4\), then
\[ 2(4) + 3y = 5 \] \[ 8 + 3y = 5 \] \[ 3y = -3 \] \[ y = -1 \] So another point is \((4,-1)\).

Step 4: Find points for the second line.}
For the equation \(5x - 4y = -22\):
If \(x = -2\), then
\[ 5(-2) - 4y = -22 \] \[ -10 - 4y = -22 \] \[ -4y = -12 \] \[ y = 3 \] So one point is \((-2,3)\).
If \(x = 2\), then
\[ 5(2) - 4y = -22 \] \[ 10 - 4y = -22 \] \[ -4y = -32 \] \[ y = 8 \] So another point is \((2,8)\).

Step 5: Find the intersection point algebraically for confirmation.}
From the first equation:
\[ 2x + 3y = 5 \] From the second equation:
\[ 5x - 4y = -22 \] Multiply the first equation by \(4\):
\[ 8x + 12y = 20 \] Multiply the second equation by \(3\):
\[ 15x - 12y = -66 \] Now add both equations:
\[ 23x = -46 \] \[ x = -2 \] Substitute \(x = -2\) into \(2x + 3y = 5\):
\[ 2(-2) + 3y = 5 \] \[ -4 + 3y = 5 \] \[ 3y = 9 \] \[ y = 3 \]
Step 6: State the graphical solution.}
Thus, the two lines intersect at the point:
\[ (-2,3) \] Hence, the graphical solution is:
\[ \boxed{x = -2,\; y = 3} \]