Use substitution to solve each of the following pairs of simultaneous equations for x and y in terms of a and b:
a) ax + y = b
y = bx
b) ax + by = b
x = by
I'll do (a). #b is done the same way.
since y = bx, we substitute that into
ax + y = b
and we then have
ax + bx = b
(a+b)x = b
x = b/(a+b)
Then, since y = bx,
y = b * b/(a+b) = b^2/(a+b)
a) To solve the first pair of simultaneous equations using substitution, we will substitute the value of y from equation 2 into equation 1.
a) ax + y = b
y = bx
Substitute the value of y from equation 2 into equation 1:
ax + bx = b
Now, we can factor out x:
x(a + b) = b
To isolate x, divide both sides by (a + b):
x = b / (a + b)
So, the solution for x in terms of a and b is x = b / (a + b).
To find y, substitute the value of x into either of the original equations:
y = bx
From equation 2, substitute the value of x:
y = b * (b / (a + b))
Simplify the expression:
y = b^2 / (a + b)
Therefore, the solution for y in terms of a and b is y = b^2 / (a + b).
b) To solve the second pair of simultaneous equations using substitution, we will substitute the value of x from equation 2 into equation 1.
ax + by = b
x = by
Substitute the value of x from equation 2 into equation 1:
a(by) + by = b
Now, we can factor out y:
(ab + 1)y = b
To isolate y, divide both sides by (ab + 1):
y = b / (ab + 1)
So, the solution for y in terms of a and b is y = b / (ab + 1).
To find x, substitute the value of y into either of the original equations:
x = by
From equation 2, substitute the value of y:
x = b * (b / (ab + 1))
Simplify the expression:
x = b^2 / (ab + 1)
Therefore, the solution for x in terms of a and b is x = b^2 / (ab + 1).
In both cases, we have found the solutions for x and y in terms of a and b using the method of substitution.