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Chapter 1: Problem 108

Given \(y=3 x-2(5-x),\) find the value of \(x\) for which \(y=0 .[1.1]\)

Short Answer

Expert verified
The value of \(x\) that makes \(y = 0\) in the given equation is \(x = 2\).

Step by step solution

01

Substitute Y in the equation with 0

Replace \(y\) with 0 in the given formula. This results in the equation \(0 = 3x - 2(5-x)\).
02

Simplify the equation

Expand the brackets to get \(0 = 3x - 10 + 2x\). Then, combine like terms to get the simplified equation: \(0 = 5x - 10\).
03

Solve for X

This equation can now be solved for \(x\). First, add 10 to both sides of the equation resulting in \(10 = 5x\). Lastly, dividing both sides by 5 will give you the value of \(x\), which is \(x = 10/5 = 2\).

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Key Concepts

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

Equation Simplification
When tackling algebra problems, one of the first crucial steps is simplifying the equation. This involves making the equation as straightforward as possible while maintaining its balance. Let's break down what this process typically includes:
  • Removing Parentheses: In our given equation, we initially see several terms involving parentheses. The first task is to expand these, focusing on multiplication within the parentheses.
  • Calculating Constants: Constants are numbers without variables. Simplifying may involve arithmetic on these numbers to further condense the equation.
  • Arranging Terms: By organizing the terms in a more logical manner, it sets the stage for combining them, leading us into the next core concept.
Effective simplification ensures that we maintain all mathematical relationships in the equation. It prepares us for solving unknown variables with a clearer path.
Substitution Method
The substitution method is a powerful tool in algebra for solving equations. It involves replacing variables with known values or variables from other equations. In this case, we took the given equation with known values for certain terms:
  • Replacing Known Values: Here, the problem asks us to find the value of \(x\) for which \(y = 0\). By substituting \(y\) with zero, our job becomes focused on solving for \(x\).
  • Establishing Equation for Single Variable: Once \(y\) is replaced, the equation transitions into one solely based on \(x\), simplifying the process significantly.
This method refines equations into simpler forms that are often much easier to handle, especially when finding solutions for specific variables.
Combining Like Terms
In algebra, combining like terms is the process of simplifying an equation by adding or subtracting terms that have the same variables raised to the same power. Consider this as tidying up the equation by grouping similar elements:
  • Identify Like Terms: Look for terms that have the same variable part. In the equation \(3x + 2x\), both terms are like terms because they contain \(x\).
  • Sum or Subtract the Coefficients: Once identified, add or subtract their coefficients. In our solution, \(3x\) combined with \(2x\) yields \(5x\).
Combining like terms is similar to organizing clutter by categorizing similar items together. This consolidation reduces complexity and aids in isolating variables for solving equations.

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