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Chapter 3: Problem 56

If \(6 a+b=12\) and \(4 a+2 b=0\), what is \(b / a\) ? (A) \(-4\) (B) 2 (C) \(-2\) (D) \(-1 / 2\) (E) \(-1 \frac{1}{2}\)

Short Answer

Expert verified
\(-2\)

Step by step solution

01

- Write down the given equations

Start by recording the provided equations:1) \[6a + b = 12\]2) \[4a + 2b = 0\]
02

- Solve the second equation for b

Simplify the second equation to isolate b:\[4a + 2b = 0\]Divide every term by 2:\[2a + b = 0\]Rearrange to solve for b:\[b = -2a\]
03

- Substitute b into the first equation

Replace b in the first equation with the expression found in Step 2:\[6a + (-2a) = 12\]This simplifies to:\[4a = 12\]
04

- Solve for a

Divide both sides by 4:\[a = 3\]
05

- Substitute a back into b = -2a

Using the value of a found in Step 4, calculate b:\[b = -2(3) = -6\]
06

- Find the ratio b / a

Substitute the values of a and b found:\[\frac{b}{a} = \frac{-6}{3} = -2\]
07

- Choose the correct answer

From the given options, the correct answer is (C) \[-2\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Linear Equations
Linear equations are the building blocks of algebra. They represent straight lines when graphed and usually take the form of either one variable or two-variable equations. In the example exercise, we are given two linear equations involving two variables, a and b:
\[6a + b = 12\] and \[4a + 2b = 0\].
To solve these, we need to find the values of a and b that satisfy both equations at the same time. This process is crucial for problems requiring you to find specific quantities or relationships between variables. Linear equations can be solved using different methods such as substitution or elimination, which will be discussed in the following sections.
Ratios
A ratio is a comparison between two quantities. It indicates how many times one number contains another and is often written as a fraction. In the given exercise, we need to find the ratio \[ \frac{b}{a}\].
Here are a few steps to understand ratios better:
  • A ratio of 2:1 means that for every 2 units of the first quantity, there is 1 unit of the second quantity.
  • Ratios can be simplified just like fractions by dividing both terms by their greatest common divisor (GCD).
  • Ratios can help to compare and understand relationships between different quantities in a problem.

In our problem, once we found the exact values of a and b, we expressed the ratio by simply dividing them. This step is crucial in many GMAT problems where the relationship between quantities needs to be clearly defined.
Substitution Method
The substitution method is an algebraic tool used to solve systems of equations. It involves solving one equation for one variable and then plugging that expression into the other equation.
Let's break down the process:
  • Start with one of the given equations and isolate one variable. In our example, from \[4a + 2b = 0\], we isolated b to get \[b = -2a\].
  • Next, substitute this expression into the other equation to eliminate one variable. This step converts the system into a single-variable equation, which can then be solved straightforwardly. In our case, we replaced b in \[6a + b = 12\] to get \[6a + (-2a) = 12\].
  • This simplified to \[4a = 12\], making it easy to solve for a.

After finding the value of a, we used it to calculate b, ensuring both values satisfy the original equations. This method is reliable and often easier than elimination, especially when one variable is easily isolated.

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