Answer:
a) [tex]a = \dfrac{c}{r - 2}[/tex]
b)[tex]a = \dfrac{4 - y}{3x + y}[/tex]
d) [tex] a = \dfrac{15}{db + 10} [/tex]
e) [tex] a = \dfrac{3b + c^2}{1 - 3b} [/tex]
Step-by-step explanation:
Let's rearrange each equation to isolate [tex]a[/tex] as the subject.
a) Given: [tex]r = \dfrac{2a+c}{a}[/tex]
Multiply both sides by [tex]a[/tex]:
[tex]r \cdot a = 2a + c[/tex]
Now, subtract [tex]2a[/tex] from both sides:
[tex]r \cdot a - 2a = c[/tex]
Factor out [tex]a[/tex] on the left side:
[tex]a(r - 2) = c[/tex]
Finally, divide both sides by [tex]r - 2[/tex]:
[tex]a = \dfrac{c}{r - 2}[/tex]
[tex]\dotfill[/tex]
b) Given: [tex]y = \dfrac{4 - 3xa}{1 + a}[/tex]
Multiply both sides by [tex]1 + a[/tex]:
[tex]y(1 + a) = 4 - 3xa[/tex]
Expand the left side:
[tex]y + ya = 4 - 3xa[/tex]
Now, move the term [tex]3xa[/tex] to the left side:
[tex]3xa + ya = 4 - y[/tex]
Factor out [tex]a[/tex] from the left side:
[tex]a(3x + y) = 4 - y[/tex]
Divide both sides by [tex]3x + y[/tex]:
[tex]a = \dfrac{4 - y}{3x + y}[/tex]
[tex]\dotfill[/tex]
d) Given: [tex]d = \dfrac{5(3-2a)}{ba}[/tex]
First, clear the fraction by multiplying both sides by [tex]ba[/tex]:
[tex] d \times ba = 5(3-2a) [/tex]
Expand the right side:
[tex] dba = 15 - 10a [/tex]
Move the term with [tex]a[/tex] to one side by adding [tex]10a[/tex] to both sides:
[tex] dba + 10a = 15 [/tex]
Factor out [tex]a[/tex] from the left side:
[tex] a(db + 10) = 15 [/tex]
Finally, solve for [tex]a[/tex] by dividing both sides by [tex](db + 10)[/tex]:
[tex] a = \dfrac{15}{db + 10} [/tex]
[tex]\dotfill[/tex]
e) Given: [tex] b = \dfrac{a - c^2}{3(a+1)} [/tex]
Multiply both sides of the equation by [tex] 3(a + 1) [/tex] to eliminate the denominator:
[tex] b \times 3(a + 1) = a - c^2 [/tex]
[tex] 3b(a + 1) = a - c^2 [/tex]
Expand the left side:
[tex] 3ab + 3b = a - c^2 [/tex]
Move all terms involving [tex] a [/tex] to one side and other terms to the other side:
[tex] a - 3ab = 3b + c^2 [/tex]
Factor out [tex] a [/tex] on the left side:
[tex] a(1 - 3b) = 3b + c^2 [/tex]
Divide both sides by [tex] (1 - 3b) [/tex] to isolate [tex] a [/tex]:
[tex] a = \dfrac{3b + c^2}{1 - 3b} [/tex]
