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Chapter 2: Problem 63

Solve each equation, and check the solution. \(4 x-6-3 x=3\)

Short Answer

Expert verified
x = 9

Step by step solution

01

Combine like terms

Combine the terms on the left side of the equation that contain the variable x: \[4x - 6 - 3x = 3\]Combining the x terms, we get: \[(4x - 3x) - 6 = 3\]This simplifies to: \[x - 6 = 3\]
02

Isolate the variable

To isolate x, add 6 to both sides of the equation: \[x - 6 + 6 = 3 + 6\]Which simplifies to: \[x = 9\]
03

Check the solution

Substitute x back into the original equation to ensure the solution is correct: \[4(9) - 6 - 3(9) = 3\]First, calculate the products: \[36 - 6 - 27 = 3\]Then combine the constants: \[36 - 33 = 3\]This simplifies to: \[3 = 3\]Since both sides of the equation are equal, the solution is verified.

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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 of the first steps is combining like terms. Like terms are terms that contain the same variable raised to the same power.
For example, in the equation: \[4x - 6 - 3x = 3\]the terms \[4x\] and \[3x\] are like terms because they both contain the variable \[x\] raised to the first power. Combining these terms results in:\[(4x - 3x) - 6 = 3\]
This simplifies to \[x - 6 = 3\].
By combining like terms, we reduce the complexity of the equation and simplify further steps.
isolating variable
After combining like terms, the next step is to isolate the variable. Isolating the variable means getting the variable by itself on one side of the equation.
For the simplified equation \[x - 6 = 3\], we need to get \[x\] on its own:
To do so, we add 6 to both sides of the equation:
\[x - 6 + 6 = 3 + 6\]
This simplifies to:
\[x = 9\]
Adding or subtracting terms to both sides is a common way to isolate the variable in linear equations.
checking solutions
Once we find a solution, it's crucial to verify its correctness by substituting it back into the original equation. This step ensures there were no mistakes made during previous steps.
In our case, the solution is \[x = 9\]. Substituting \[x\] back into the original problem results in:
\[4(9) - 6 - 3(9) = 3\]
First, calculate the products:
\[36 - 6 - 27 = 3\]
Then combine the constants to simplify:
\[36 - 33 = 3\]
Since both sides of the equation match, we confirm that our solution \[x = 9\] is correct. By checking solutions, we ensure the accuracy and completeness of our problem-solving process.

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Most popular questions from this chapter

The world's largest taco contained approximately \(1 \mathrm{~kg}\) of onion for every \(6.6 \mathrm{~kg}\) of grilled steak. The total weight of these two ingredients was \(617.6 \mathrm{~kg}\). To the nearest tenth of a kilogram, how many kilograms of each ingredient were used to make the taco? (Data from Guinness World Records.

If four times a number is added to \(8,\) the result is equal to three times the number, added to \(5 .\) Find the number.

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