Solve:
3x + 7y + 5= 0
4x - 3y - 8 = 0
Find p; if:
px + y - 1= 0
3x+7y=-5 (i)
4x-3y=8 (ii)
(i)x 4:
12x+28y=-20 (iii)
(ii)x 3:
12x-9y=24 (iv)
(iv)-(iii):
-37y=44
y=-44/37
subs y into (i):
3x+7(-44/37)=-5
3x= -5+308/37
x=148/37
3x = -185/37 + 308/37 = 123/37
x = 41/37
To solve the system of equations:
3x + 7y + 5 = 0 ... (Equation 1)
4x - 3y - 8 = 0 ... (Equation 2)
Let's first eliminate one variable by multiplying one of the equations by a constant so that the coefficients of either x or y will cancel out when added with the other equation.
Let's start by eliminating y.
Multiply Equation 1 by 3 and Equation 2 by 7:
(3)(3x + 7y + 5) = 0 becomes 9x + 21y + 15 = 0 ... (Equation 3)
(7)(4x - 3y - 8) = 0 becomes 28x - 21y - 56 = 0 ... (Equation 4)
Now, add Equation 3 and Equation 4:
(9x + 21y + 15) + (28x - 21y - 56) = 0
Combine like terms:
9x + 28x + 21y - 21y + 15 - 56 = 0
37x - 41 = 0
Now, isolate x:
37x = 41
x = 41/37
x = 1.1081 (rounded to four decimal places)
Next, substitute the value of x back into one of the original equations to solve for y. Let's use Equation 1:
3x + 7y + 5 = 0
Plug in x = 1.1081:
3(1.1081) + 7y + 5 = 0
Simplify:
3.3243 + 7y + 5 = 0
Combine like terms:
7y + 8.3243 = 0
Isolate y:
7y = -8.3243
y = -8.3243/7
y = -1.1892 (rounded to four decimal places)
Now, let's move on to the second part of the question:
We need to find p if:
px + y - 1 = 0
Substitute the values of x = 1.1081 and y = -1.1892:
p(1.1081) + (-1.1892) - 1 = 0
Simplify:
1.1081p - 1.1892 - 1 = 0
Combine like terms:
1.1081p - 2.1892 = 0
Add 2.1892 to both sides:
1.1081p = 2.1892
Now, isolate p:
p = 2.1892 / 1.1081
p = 1.9731 (rounded to four decimal places)
Therefore, the value of p is approximately 1.9731.