Solving for 'x' in terms of 'y' type of problem

Need help with these 2, if you could help step-by-step that's really helpful!

Solve for v in terms of w:

2v + 7 = w - 3

----------------------------------

Solve for s in terms of r:

4r + 5s + 2 = 8r - s + 7

Quote:

Originally Posted by Sarah-

Need help with these 2, if you could help step-by-step that's really helpful!

Solve for v in terms of w:

2v + 7 = w - 3

subtract 7 from both sides, then divide both sides by 2

----------------------------------

Solve for s in terms of r:

4r + 5s + 2 = 8r - s + 7

add s to both sides, subtract 4r from both sides, subtract 2 from both sides ... lastly, divide both sides by the resulting coefficient of s.

...

Quote:

Originally Posted by Sarah-

Solve for v in terms of w:
2v + 7 = w - 3

$2v + 7 = w - 3 \implies 2v = w-10 \implies v = \boxed{\frac{w-10}{2}}$
STEPS:

Spoiler:

The key:

Spoiler:

Our goal is to have v on one side and the rest on the other side of the equation. The key is to keep the balance, what-ever we add/divide/multiply/subtract on one side, we /divide/multiply/subtract on the other side as well. This is because, say, if you had five coins on one hand and five on the other, then you can say that the number of coins in your left hand is equal to the number of coins in your right hand (that is what the ''='' sign means); but if you take one of out of the right hand and nothing on the left, then what you have said about your left-hand coins and right-hand being equal no longer holds (and the the ''='' sign no longer applies)). That understood, the rest is easy.

$2v+7 = w-3.$

Subtract $7$ from both sides:

$2v = w-3-7$

$2v = w-10$

Divide both sides by $2$ :

$\boxed{v = \frac{w-10}{2}}<br />$

Quote:

Originally Posted by Sarah-

Solve for s in terms of r:

4r + 5s + 2 = 8r - s + 7

$4r + 5s + 2 = 8r - s + 7 \implies 6s = 4r+5 \implies \boxed{s = \frac{4r+5}{6}}$
STEPS:

Spoiler:

The key (again):

Spoiler:

Our goal is to have v on one side and the rest on the other side of the equation. The key is to keep the balance, what-ever we add/divide/multiply/subtract on one side, we /divide/multiply/subtract on the other side as well. This is because, say, if you had five coins on one hand and five on the other, then you can say that the number of coins in your left hand is equal to the number of coins in your right hand (that is what the ''='' sign means); but if you take one of out of the right hand and nothing on the left, then what you have said about your left-hand coins and right-hand being equal no longer holds (and the ''='' sign no longer applies). That understood, the rest is easy.

$4r+5s+2 = 8r-s+7$

Subtract $2$ from both sides:

$4r+5s+2-2= 8r-s+7-2$

$4r+5s = 8r-s+5$

Subtract 4r from both sides:

$4r-4r+5s= 8r-4r-s+5$

$5s = 4r-s+5$

Add s to both sides:

$5s+s = 4r-s+s+5$

$6s = 4r+5$

Divide both sides by $6$ :

$\frac{6s}{6} = \frac{4r+5}{6}$

Then you have the answer:

$s = \frac{4r+5}{6}$