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Chapter 3: Problem 61

Solve the equation. $$-3 x+7+4 x=12$$

Short Answer

Expert verified
x = 5

Step by step solution

01

Combine Like Terms

First, combine the like terms on the left side of the equation. The terms -3x and 4x are like terms. -3x + 4x + 7 = 12
02

Simplify the Combined Terms

Simplify the expression by combining the coefficients of x. (-3 + 4)x + 7 = 12 This results in: x + 7 = 12
03

Isolate the Variable

Subtract 7 from both sides to isolate the variable x. x + 7 - 7 = 12 - 7 Simplifying this, we get: x = 5

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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 to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, in the equation \( -3x + 7 + 4x = 12 \), -3x and 4x are both terms that contain the variable x. To combine them, you simply add or subtract their coefficients:
  • -3 + 4 = 1

So, -3x + 4x becomes 1x, or simply x.
Combining like terms helps to simplify the equation, making it easier to solve.
Simplifying Expressions
Once like terms are combined, the next step is simplifying the expression. Simplification involves performing basic arithmetic operations to reduce the equation to its simplest form.
For instance, in the equation \( x + 7 = 12 \), we can see that it is already simplified as much as possible. Simplifying also involves distributing coefficients, consolidating constants, and eliminating unnecessary terms.
Simplifying expressions not only makes the equation easier to solve but also helps in reducing potential errors.
Isolating Variables
To find the value of a variable, it must be isolated on one side of the equation. This involves moving other terms to the opposite side.
In the equation \( x + 7 = 12 \), we need to isolate x. To do this, subtract 7 from both sides:
  • x + 7 - 7 = 12 - 7

Performing the subtraction gives us:
\( x = 5 \).
Isolating the variable is vital in finding its value, as it leaves us with a straightforward answer.
Prealgebra
Prealgebra is the foundation for all future algebra and math courses. It involves understanding basic mathematical concepts and operations. In prealgebra, you learn how to:
  • Combine like terms
  • Simplify expressions
  • Isolate variables
  • Work with basic arithmetic operations

Mastering prealgebra concepts is crucial, as it prepares you for more advanced topics like algebra, geometry, and calculus.
With a strong prealgebra foundation, solving linear equations and other mathematical problems becomes much more manageable.

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