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Chapter 3: Problem 62

Solve the equation. $$6 x+7-5 x=10$$

Short Answer

Expert verified
x = 3

Step by step solution

01

Combine like terms

Look at the terms on the left side of the equation that contain the variable x. Combine the terms to simplify the equation:\(6x - 5x + 7 = 10\)This simplifies to:\(x + 7 = 10\)
02

Isolate the variable x

Subtract 7 from both sides of the equation to isolate x:\(x + 7 - 7 = 10 - 7\)Which simplifies to:\(x = 3\)

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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, the first step is often combining like terms. Like terms are terms that contain the same variable raised to the same power. For example, in the equation:

6x + 7 - 5x = 10

Both 6x and -5x are like terms because they both contain the variable 'x'. To combine them, you simply add or subtract their coefficients. Here: 6x - 5x = x

This simplifies our equation to:
x + 7 = 10

Combining like terms helps to simplify the equation, so it is easier to solve. It reduces the number of terms which makes the next steps more straightforward.
Isolating the Variable
The goal of solving an equation is to find the value of the variable. To do this, you need to isolate the variable on one side of the equation. For example, after combining like terms, our equation is:

x + 7 = 10

To isolate 'x', you need to get rid of the '+7'. You can do this by performing the opposite operation. Since '+7' is added to x, you need to subtract 7 from both sides of the equation:

x + 7 - 7 = 10 - 7

This results in:
x = 3

Now, the variable is isolated. You have x alone on one side of the equation and the solution (3) on the other side.
Basic Algebra
Understanding basic algebra is crucial for solving equations. Algebra involves working with variables (like 'x') which stand in for unknown values. The primary goal is to solve for these unknowns. Basic algebra includes operations such as:

  • Combining like terms
  • Performing arithmetic operations
  • Using properties of equality to simplify equations

In the context of our example, basic algebra allows us to use the properties of equality (like adding or subtracting the same value from both sides of the equation) to isolate the variable and find its value. Learning these fundamentals makes it easier to handle more complex algebraic problems later on.
Equation Simplification
Simplifying an equation is about making it easier to solve by performing operations like combining like terms and isolating the variable. It involves reducing the equation to a simpler form without changing its solutions. For example:

The original equation:
6x + 7 - 5x = 10

After combining like terms:
x + 7 = 10

After isolating the variable:
x = 3

By systematically simplifying the equation step-by-step, we make it easier to identify the value of the variable. Simplification helps prevent mistakes and ensures you understand the problem better. It's a foundational skill in algebra that will be useful in more complex problems.

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