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Chapter 5: Problem 15

If \(a-b=-3\) and \(a+b=11\), then: A. \(a b=8\) B. \(a / b=7 / 3\) C. \(a b=21\) D. \(a / b=7 / 4\) E. \(a b=28\)

Short Answer

Expert verified
E. \(ab=28\)

Step by step solution

01

- Solve for 'a'

Given the equations: 1) \(a - b = -3\) 2) \(a + b = 11\)First, add both equations to eliminate 'b'.\[(a - b) + (a + b) = -3 + 11\]Simplify to get:\[2a = 8\]Divide by 2:\[a = 4\]
02

- Solve for 'b'

Now substitute the value of 'a' back into one of the original equations. Use the second equation for simplicity:\[4 + b = 11\]Solve for 'b':\[b = 7\]
03

- Verify the values and calculate

Use the values of 'a' and 'b' to check the expressions given in the options:\[a = 4, b = 7\]Calculate the products and ratios:1) \(ab = 4 \times 7 = 28\)2) \(\frac{a}{b} = \frac{4}{7}\)Compare these with the given options.

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Key Concepts

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

Simultaneous Equations
To solve problems involving simultaneous equations, you need to find values for variables that satisfy all given equations simultaneously. In our example, we have two equations:
  • \(a - b = -3\)
  • \(a + b = 11\)
The goal is to find values of 'a' and 'b' that work for both equations. This usually involves methods like substitution or elimination. In this exercise, we use the elimination method. Adding these two equations helps us eliminate 'b' and easily solve for 'a'. This step reduces the complexity of the problem and makes finding the variables straightforward.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations to isolate variables. In our simultaneous equations problem, we added the two equations to eliminate one variable. Here's how:
  • Add both equations: \((a - b) + (a + b) = -3 + 11\)
  • This simplifies to \(2a = 8\)
  • Divide by 2 to get \(a = 4\)
Next, we substitute 'a' back into one of the original equations to find 'b'. In this case:
  • Use the second equation: \(4 + b = 11\)
  • Solve for 'b': \(b = 7\)
Mastering these algebraic manipulations is crucial for the GMAT math section.
GMAT Math Section
The GMAT math section tests your problem-solving abilities, especially in areas like algebra, arithmetic, and geometry. Knowing how to work with simultaneous equations and manipulate algebraic expressions is essential.
The section often includes:
  • Problem-solving questions like the one in this exercise
  • Data sufficiency questions, which assess your ability to determine whether you have enough information to solve a problem
Fast and accurate algebraic manipulation is key to doing well, as it saves time and reduces errors.
Verification of Solutions
Verification of solutions means checking that your computed values actually satisfy the original equations and the given options. For our example, after finding \(a = 4\) and \(b = 7\), we need to verify these values:
  • Original equations:
    • \(4 - 7 = -3\) (True)
    • \(4 + 7 = 11\) (True)
  • Given options:
    • \(ab = 4 \times 7 = 28\) (Matches option E)
Verification ensures that your solutions are correct and align with the given options.

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