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Chapter 2: Problem 58

\(3-6 x=-7 x+10\)

Short Answer

Expert verified
x = 7

Step by step solution

01

Move variables to one side of the equation

To isolate the variable, add \(7x\) to both sides of the equation: \(3 - 6x + 7x = 10\)
02

Simplify the equation

Combine like terms on the left side: \(3 + x = 10\)
03

Isolate x

Subtract 3 from both sides to solve for \(x\): \(x = 10 - 3\)
04

Simplify the result

Subtract the numbers to find the value of \(x\): \(x = 7\)

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Key Concepts

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

Isolating the variable
One of the first steps in solving linear equations is isolating the variable. This means getting the variable you are solving for, such as x, on one side of the equation.

Let's look at our equation: 3 - 6x = -7x + 10.
We need to move all terms containing x to one side.
Add 7x to both sides to start: 3 - 6x + 7x = -7x + 10 + 7x.
Now the equation is: 3 + x = 10, which we will simplify in the next steps.
Combining like terms
Combining like terms simplifies the equation, making it easier to solve.
In our example: 3 - 6x + 7x = 10, the left side of the equation has -6x + 7x.
These are like terms because they both have x. Simply add them together: -6x + 7x = x.
Now, our equation becomes: 3 + x = 10.
It's easier to see that we're closer to solving for x.
Subtracting terms
Subtracting terms is essential in moving constants to one side of the equation.
In our simplified equation: 3 + x = 10, we need to isolate x by removing the constant 3.
Subtract 3 from both sides: 3 + x - 3 = 10 - 3.
This removes the 3 from the left side: x = 7.
Now x stands alone, and we have found its value.
Simplifying equations
Simplifying equations is a key process in solving for the variable.
It involves making the equation simpler at each step.
In our case: 3 - 6x = -7x + 10, we went through these steps:
  • Isolating the variable by moving terms
  • Combining like terms
  • Subtracting constants

Each action made the equation simpler: 3 + x = 10 became x = 7.
The final simplified result tells us the value of x.

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