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Chapter 3: Problem 59

Solve each equation. \(5-5=3 x+2 x\)

Short Answer

Expert verified
The solution to the equation is \(x = 0\).

Step by step solution

01

Simplify the Equation

Begin by simplifying the left side of the equation. Calculate the subtraction on the left side: \(5 - 5 = 0\). The equation now becomes \(0 = 3x + 2x\).
02

Combine Like Terms

Combine the \(x\) terms on the right side of the equation. Add \(3x + 2x\) to get \(5x\). The equation now is \(0 = 5x\).
03

Solve for x

To isolate \(x\), divide both sides of the equation by 5. This results in \(x = 0\).

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Key Concepts

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

Combining Like Terms
When solving linear equations, one important step is combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, in the expression \(3x + 2x\), both terms have the variable \(x\) raised to the first power. Therefore, they are considered like terms. By combining them, we simplify the equation and make it easier to solve.

To combine like terms:
  • Add or subtract the coefficients of the like terms.
  • Keep the variable part of the term unchanged.
In our case, adding \(3x\) and \(2x\) gives us \(5x\). Combining like terms can streamline complex equations, making the problem more manageable.
Simplifying Equations
Simplifying equations is a process used to make equations easier to solve. It involves making the expression on each side of the equation as simple as possible. The goal is to reduce the equation to its most basic form.

Here are some standard techniques for simplifying equations:
  • Perform basic arithmetic operations like addition, subtraction, multiplication, or division as needed. For example, on the left side of the equation, we performed the subtraction \(5 - 5\) to get \(0\).
  • Combine like terms to reduce variables on one side of the equation.
  • Cancel out terms that appear on both sides of an equation when applicable.
By simplifying, you make it easier to spot the steps needed to solve the equation.
Dividing to Isolate Variables
After you have simplified an equation and combined like terms, the next crucial step is isolating the variable. Isolating the variable involves getting the variable \(x\) by itself on one side of the equation.

This often involves using division, especially when the variable is multiplied by a coefficient. In our example, we have the equation \(0 = 5x\). To solve for \(x\), we need to divide both sides by 5.

Steps for isolating a variable by division:
  • Look for the coefficient of the variable term.
  • Divide each side of the equation by this coefficient.
  • After division, you should have the variable by itself: \(x = 0\).
Dividing to isolate variables is a fundamental part of finding the solution to linear equations. It helps provide the numerical value for the variable.

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