# Algebra: Substitution

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## 5/30 Algebra: Substitution

Always follow the Knowsys Method for math problems. It will save you time not only on the SAT, but also on math tests and quizzes in your school classes. Read the question carefully and identify the bottom line. Assess your options. Ask what you could do and then what you should do, and solve the problem quickly and efficiently once you have decided on a strategy. Finally, loop back to double check that you answered the question correctly

If A,B, and C are numbers such that  and , then  is equal to which of the following?

At first this problem may seem to be challenging but it is important that we tackle it step by step. Most of the more difficult problems on the SAT consist of 3 or 4 steps, each of which is fairly easy on their own. As long as we focus on the next step and don't panic, most of the "hard" problems will actually turn out to be fairly straightforward. The first step is always to identify the "bottom line". In this problem, we are looking for what  is equal to.
Since the equation currently has 3 different variables we cannot solve it as it is. In order to eliminate some of the variables we will need to use substitution. We start with first equation

$\frac{a}{b}=3$

We can manipulate this equation to give us

$a=3b$.

We then substitute this into the original equation to get

$\frac{a+b}{b+c} = \frac{3b+b}{b+c}=\frac{4b}{b+c}$

Now we work on eliminating c. By solving the second equation we get the following.

$\frac{b}{c}=7 \therefore \frac{b}{7}=c$

Substituting this into our equation now gives us

$\frac{4b}{b+c}=\frac{4b}{b+\frac{b}{7}}=\frac{4b}{\frac{8b}{7}}$

At this point, you may be starting to panic because the equations look so complicated. If so, take a deep breath and relax, we are almost done. Remember that with fractions, dividing is the same as multiplying by a reciprocal. We can now solve our equation since we only have one variable.

$\frac{4b}{\frac{8b}{7}} = \frac{4b}{1}\times \frac{7^{}}{8b}=\frac{28b}{8b}=\frac{7}{2}$

Now all that's left to do is look at the answers and find which one matches our solution.

(A)
(B)
(C)
(D)
(E)

The answer is A and that matches our prediction exactly!

On sat.collegeboard.org, 42% of the responses were correct.

For more help with math, visit www.myknowsys.com!