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Chapter 2: Problem 57

Use the intersection-of-graphs method to solve the equation. Then solve symbolically. 2(x - 1) - 2 = x

Short Answer

Expert verified
The solution is \( x = 4 \).

Step by step solution

01

Rewrite the Equation

Start by rewriting the given equation to make it clear for solving: \[ 2(x - 1) - 2 = x \].This will be the equation we will use for both graphing and symbolic methods.
02

Simplify the Equation

Distribute the 2 in the left side of the equation and simplify:\[ 2x - 2 - 2 = x \].This simplifies to:\[ 2x - 4 = x \].
03

Rearrange for Intersection Points

Move all terms to one side to prepare for symbolic solving:\[ 2x - 4 - x = 0 \].This simplifies to:\[ x - 4 = 0 \].
04

Solve Symbolically

Add 4 to both sides to solve for \( x \):\[ x = 4 \].This is the symbolic solution of the equation.
05

Graph Each Side of Equation

Consider the left side of the original equation as \( y_1 = 2(x - 1) - 2 \) and the right side as \( y_2 = x \).Graph \( y_1 = 2x - 4 \) and \( y_2 = x \) separately.
06

Find Intersection on Graph

On the graph, observe that the two lines \( y_1 = 2x - 4 \) and \( y_2 = x \) intersect at the point where \( x = 4 \).This confirms the solution found symbolically.

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Key Concepts

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

Intersection of Graphs Method
The intersection-of-graphs method is a visual approach to solving equations. It helps you find the solution by identifying where two graphs meet or intersect. To use this method:
  • Rewrite your equation so that each side represents a different function. For example, transforming the given equation into two functions: one for the left side and one for the right side.
  • Consider the equations: \( y_1 = 2(x - 1) - 2 \) and \( y_2 = x \). By plotting these equations as graphs on the same set of axes, you establish a visual empowerment.
  • The solution to the equation is represented by the x-coordinate where the two graphs intersect. In this example, both graphs intersect at the point \( x = 4 \).
This method is helpful for visual learners and can be used for different types of equations, especially when algebraic methods become too complex or when a quick estimate is necessary.
Symbolic Solving
Symbolic solving is a method that employs algebraic manipulation to solve equations. It's a step-by-step process where you aim to find the value of the unknown variable. Let's break it down:
  • Start with the equation, \( 2(x - 1) - 2 = x \), and simplify it by distributing and combining like terms.
  • The simplification process results in \( 2x - 4 = x \). The goal is to isolate \( x \) on one side of the equation for easy solving.
  • By moving all terms involving \( x \) to one side, you get \( x - 4 = 0 \).
  • Finally, solve for \( x \) by adding 4 to both sides, giving you the result: \( x = 4 \).
Symbolic solving is an essential skill for students, as it helps to understand the logic and structure behind equations, which is pivotal in more advanced math courses.
Algebraic Simplification
Algebraic simplification streamlines equations by reducing them to their simplest form. This process makes the math more manageable and the solution clearer. Here's a closer look at how it's done:
  • Begin by applying the distributive property: \( 2(x - 1) - 2 = x \) becomes \( 2x - 2 - 2 \).
  • Combine the constant terms on the left: \( 2x - 4 = x \), which is more concise and easier to manipulate.
  • Further rearrange to isolate variables. Here, subtract \( x \) from both sides to simplify the equation to \( x - 4 = 0 \).
  • Add 4 on both sides to isolate \( x \) completely, yielding \( x = 4 \).
Using algebraic simplification, you effectively make complex equations more comprehensible, allowing for better problem-solving skills and an enhanced understanding of algebraic processes.

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Most popular questions from this chapter

In 2002 sales of premium online music totaled \(\$ 1.6\) billion. In 2005 this revenue reached \(\$ 3.6\) billion. (A) Find a point-slope form of the line passing through \((2002,1.6)\) and \((2005,3.6) .\) Interpret the slope. (B) Use the equation to estimate projected sales in2008. Did you use interpolation or extrapolation? (C) Find the slope-intercept form of this line.

Solve the linear equation with the intersection-of-graphs method. Approximate the solution to the nearest thousandth whenever appropriate. $$ 8-2 x=1.6 $$

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ x-3=2 x+1 $$

Let \(y\) be directly proportional to \(x\) Complete the following. Find \(y\) when \(x=2.5,\) if \(y=13\) when \(x=10\)

Use tables to solve the equation numerically to the nearest tenth. $$ 1-6 x=7 $$
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