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Chapter 2: Problem 314

Using your own words, list the steps in the general strategy for solving linear equations.

Short Answer

Expert verified
Simplify both sides, move variables to one side, isolate the variable, and check the solution.

Step by step solution

01

- Simplify Both Sides

Combine like terms and simplify each side of the equation separately. This means adding or subtracting any constants and combining any similar variable terms.
02

- Move Variables to One Side

Use addition or subtraction to get all variable terms on one side of the equation and all constant (number) terms on the other side.
03

- Isolate the Variable

Use multiplication or division to solve for the variable. This often involves dividing or multiplying both sides of the equation by a number to get the variable by itself.
04

- Check Your Solution

Substitute your solution back into the original equation to ensure that it satisfies the equation. This confirms that you鈥檝e solved it correctly.

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Key Concepts

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

simplifying equations
To solve a linear equation, the first step is often to simplify both sides. This means making each side of the equation as straightforward as possible. Simplifying involves:
  • Combining like terms: Add or subtract terms that have the same variable.
  • Performing any straightforward arithmetic: Simplify any constants by adding or subtracting them.
For example, consider the equation:
\(3x + 5 - 2x = 10 + 1\)
We combine the like terms on the left side to get:
\(x + 5 = 11\)
Once each side is in its simplest form, you can move on to the next steps.
isolating variables
After simplifying both sides, the next step is to isolate the variable. This means getting the variable on one side of the equation by itself. You do this by moving all other terms to the other side. Here鈥檚 how:
  • First, decide which side you want to keep the variable. It鈥檚 often easiest to keep it on the side it鈥檚 already on.
  • Use addition or subtraction to move constant terms to the opposite side of the equation.
Consider our simplified equation from before: \(x + 5 = 11\)
We subtract 5 from both sides to isolate the variable:
\(x = 6\)
Now, the variable is isolated, and we can solve for it directly.
checking solutions
After finding the value of the variable, you should always check your solution. This step helps ensure that your answer is correct. To check your solution:
  • Substitute the variable's value back into the original equation.
  • Simplify both sides to see if they are equal.
Using our example:
The solution we found was \(x = 6\).
Substitute this back into the original equation:
\(3(6) + 5 - 2(6) = 10 + 1\)
Simplify it to get:
\(18 + 5 - 12 = 11\),
which simplifies further to: \(11 = 11\)
Since both sides are equal, the solution is correct.
combining like terms
Combining like terms is a crucial part of simplifying equations. Like terms are terms that have the same variables raised to the same power. For example, in the expression \(3x + 2x + 5\), the terms \(3x\) and \(2x\) are like terms because they both contain the variable \(x\).
Here鈥檚 how you combine them:
  • Add or subtract the coefficients (numbers in front of the variables) while keeping the variable part the same.
Using our example: \(3x + 2x = 5x\)
For constants, just add or subtract as you normally would: \(+5\) in our complete simplified equation:
\ 5x + 5\
Combining like terms makes the equation easier to solve and helps in isolating the variable later.

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