px+qy=r 2px-qy=2r find the solution.
There are five unknowns here: p, q, r, x, y
To find the solution to the equations px + qy = r and 2px - qy = 2r, we can use a method called elimination or substitution.
Method 1: Elimination:
1. Multiply the first equation by 2 to make the coefficients of px equal in both equations:
2(px + qy) = 2r
2px + 2qy = 2r
2. Add the multiplied equations together:
(2px - qy) + (2px + 2qy) = 2r + 2r
4px + qy = 4r
3. Now we have a new equation, 4px + qy = 4r. We can rewrite this equation as:
qy = 4r - 4px
4. Substitute the value of qy from the new equation into the first equation:
px + (4r - 4px) = r
px - 4px = r - 4r
-3px = -3r
5. Divide both sides of the equation by -3 to solve for px:
px = r
6. Substitute the value of px back into the first equation:
r + qy = r
qy = 0
7. Since qy = 0, we can set q to any value and y to any value since it won't affect the equation. Let's set them to q = 1 and y = 0 for simplicity.
8. Now we can calculate the values of p, q, r, x, and y:
px = r => px = r => p = 1
qy = 0 => q × 0 = 0 => q = any value
r = r
2px - qy = 2r
2 × 1 × x - 1 × 0 = 2r => 2x = 2r => x = r
Therefore, the solution is p = 1, q = any value, r = any value, x = r, and y = 0.
Method 2: Substitution:
1. Solve one of the equations for one variable.
From the second equation, isolate px:
2px - qy = 2r
2px = qy + 2r
px = (qy + 2r) / 2
2. Substitute this value of px into the first equation:
(qy + 2r) / 2 + qy = r
Multiply through by 2 to eliminate the fraction:
qy + 2qy + 4r = 2r
3qy + 4r = 2r
3. Simplify the equation:
3qy = 2r - 4r
3qy = -2r
4. Divide through by 3 to solve for qy:
qy = (-2r) / 3
5. Now we have a value for qy. Let's substitute it back into the first equation:
px + qy = r
px + (-2r) / 3 = r
px = r - (-2r) / 3
Multiply through by 3 to eliminate the fraction:
3px = 3r - (-2r)
3px = 3r + 2r
3px = 5r
6. Now we have an equation for px. Let's solve for p by dividing through by 3:
px = 5r / 3
p = (5r / 3) / x
7. We still have qy = (-2r) / 3. Let's solve for q by dividing through by y:
qy = (-2r) / 3
q = ((-2r) / 3) / y
8. So, the solution is p = (5r / 3) / x, q = ((-2r) / 3) / y, r = any value, x = any value, and y = any value.
In both methods, we find that the solution to the equations px + qy = r and 2px - qy = 2r is p = 1, q = any value, r = any value, x = r, and y = 0.