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Chapter 5: Problem 5

If \(4 x+6 y=20\), then \(10 x+15 y\) is: A. 40 B. 50 C. 20 D. 30 E. Cannot be determined

Short Answer

Expert verified
B. 50

Step by step solution

01

Simplify the given equation

Given the equation: 4x + 6y = 20. Divide both sides by 2 to simplify it: 2 x + 3 y = 10.
02

Scale the simplified equation

To find the value of 10x + 15y, we need to scale the simplified version of the equation: By multiplying by 5: 5 (2 x + 3 y) = 5 (10) Resulting in: 10 x + 15 y = 50.

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Key Concepts

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

linear equations
In algebra, linear equations are equations between two variables that produce a straight line when graphed. These equations typically take the form of \[ax + by = c\]where
  • \(a\) and \(b\) are coefficients,
  • \(x\) and \(y\) are variables,
  • and \(c\) is a constant.
In the original exercise, the given linear equation is \(4x + 6y = 20\). Our goal is to manipulate this equation to find the value of \(10x + 15y\).Understanding linear equations is crucial because these equations are a fundamental part of algebra. They form the basis for more complex equations and are used in various real-life applications such as calculating speed, distance, and even in economics.
mathematical simplification
Mathematical simplification involves reducing equations or expressions to their simplest form while preserving their value. For the given problem, simplifying the equation \(4x + 6y = 20\) makes it easier to work with. We simplify the equation by dividing all terms by the greatest common divisor, which is 2 in this case. This step-by-step process involves:
  • Starting with the original equation: \[4x + 6y = 20\],
  • Dividing every term by 2: \[\frac{4x}{2} + \frac{6y}{2} = \frac{20}{2}\],
  • Which simplifies to: \[2x + 3y = 10\]
This simplified equation helps in making calculations more manageable, giving a clearer path to achieve the final result.
algebraic scaling
Algebraic scaling involves adjusting the scale of an entire equation by multiplying all terms by a common factor. This technique can make solving problems easier by matching the form of what you're trying to find.In the exercise, after simplifying the equation to \(2x + 3y = 10\), the next step involves scaling it to match \(10x + 15y\). We achieve this by multiplying the entire simplified equation by 5:
  • Given: \[2x + 3y = 10\],
  • Multiplying by 5: \[5(2x + 3y) = 5 * 10\],
  • Results in: \[10x + 15y = 50\]
This step demonstrates that algebraic scaling can help transform a simpler equation into a form where you can directly find the required values, ensuring consistent results with the original equation's context.

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