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Chapter 3: Problem 2

Solve each equation. $$3 x+9+4 x=30$$

Short Answer

Expert verified
\(x = 3\).

Step by step solution

01

Combine Like Terms

The given equation is \(3x + 9 + 4x = 30\). First, combine the like terms involving \(x\). This means adding \(3x\) and \(4x\) together. This gives us \(7x\). The equation is now simplified to \(7x + 9 = 30\).
02

Isolate the Variable Term

To isolate the variable term \(7x\), subtract 9 from both sides of the equation. This gives us \(7x = 30 - 9\), which simplifies to \(7x = 21\).
03

Solve for x

Divide both sides of the equation by 7 to solve for \(x\). This gives us \(x = \frac{21}{7}\), 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 working with algebraic equations, particularly linear equations, it's crucial to simplify the equation by combining like terms. Like terms are terms that have the same variable and corresponding exponent.
In our example, we are given the equation \(3x + 9 + 4x = 30\). Here, the like terms are \(3x\) and \(4x\) because they both include the variable \(x\). To combine them, add their coefficients: \(3 + 4\).
This process results in the simplified expression \(7x\), changing the entire equation to \(7x + 9 = 30\).
By simplifying equations through combining like terms, you reduce complexity, making it easier to manipulate and solve the equation in subsequent steps.
Isolating the Variable
After combining like terms in an equation, the next task is to isolate the variable. This means getting the term with the variable by itself on one side of the equation.
In the equation \(7x + 9 = 30\), our goal is to isolate \(7x\).
  • Identify which constant term(s) need to be moved to the other side of the equation. Here, it's the \(+9\).
  • Subtract 9 from both sides to maintain balance: \(7x + 9 - 9 = 30 - 9\).
The equation now simplifies to \(7x = 21\).
Isolating the variable term makes solving for the variable straightforward since you're left with a much simpler expression.
Solving Linear Equations
Finally, to find the value of the variable, you need to solve the linear equation. Linear equations take the form \(ax + b = c\) and typically lead to a solution in one step once simplified properly.
In our example, we've reached \(7x = 21\). To solve for \(x\), divide both sides by 7, eliminating the coefficient of \(x\).
  • Perform the division: \(x = \frac{21}{7}\).
  • This simplifies to \(x = 3\).
Together, these steps show how breaking down equations into simpler parts and performing arithmetic operations leads to finding the variable's value. Solving linear equations is one of the foundational skills in algebra, setting the stage for understanding more complex mathematical concepts.

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