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Chapter 4: Problem 19

If twice a number is equal to that number minus five, what is three times that number plus seventeen minus that number?

Short Answer

Expert verified
Therefore, the expression is equal to \(7\).

Step by step solution

01

Define Variables

Let x be the number we need to find. We are given that twice the number (2x) is equal to the number minus five (x - 5). We can write this equation as: 2x = x - 5
02

Solve for x

To solve the equation for x, subtract x from both sides of the equation: 2x - x = x - x - 5 This simplifies to: x = -5
03

Calculate the Expression

Now that we've found x, we can calculate the expression given: 3 times the number (3x) -> 3(-5) = -15 Add seventeen to the result and then subtract the original number (x) -> -15 + 17 - (-5) Simplify: -15 + 17 + 5 = 7
04

Check the Result

Let's check if our result makes sense: If x is -5, then twice the number is 2(-5) = -10, and the number minus five is (-5) - 5 = -10. These values are equal, so our result is the correct solution.
05

State the Final Answer

Therefore, three times the number plus seventeen minus that number is 7.

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Key Concepts

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

Solving Linear Equations
Understanding how to solve linear equations is a foundation of algebra and a frequent type of question on standardized tests like the SAT. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.

When solving linear equations, the aim is to find the value of the unknown variable. For example, in the context of the SAT problem provided, we encounter the equation 2x = x - 5. The steps involved in solving this problem include:
  • Isolating the variable on one side by performing the same operation on both sides, such as subtracting x from each side.
  • After simplifying, the equation transforms into x = -5, leading us to the solution.
Consistent practice in solving these equations not only boosts confidence but also equips students with a critical skill for algebra-related challenges.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables (like x or y), and operation symbols. It is vital to be familiar with how to handle these expressions because they are central to understanding algebra.

In the given SAT problem, we calculate an algebraic expression by inserting the previously determined value of x into the expression 3x + 17 - x. The steps taken to compute this include:
  • Multiplying 3 by the value of x (3 times -5).
  • Adding 17 and subtracting the value of x (plus seventeen minus -5).
  • The process results in -15 + 17 + 5, which simplifies to 7.
Handling algebraic expressions efficiently is about understanding and applying the order of operations, combined with the substitution of known variables.
Test Preparation Math
Preparing for mathematics sections in tests like the SAT involves a mix of strategy and solid understanding of key concepts. It's essential to study a variety of topics, including algebra, geometry, and data analysis, as each of these areas will be tested.

Strategies for effective math test preparation include:
  • Conducting a thorough review of algebraic principles, like those applied in solving the given exercise.
  • Practicing various algebra problems until the processes become second nature.
  • Developing time management skills to ensure each question is afforded appropriate time during the actual test.
  • Learning to recognize quickly what a question is asking, which helps in applying the correct mathematical concept or formula.
With persistent study and practice, students can improve their problem-solving speed and accuracy, benefiting their test performance significantly.

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