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Chapter 0: Problem 9

Solve each linear equation. $$3(x-2)+7=2(x+5)$$

Short Answer

Expert verified
The solution to the equation is \(x = 9\).

Step by step solution

01

Distribute Multiplication into Parentheses

Distribute the coefficients 3 and 2 into their respective parentheses. This gives: \(3x - 6 + 7 = 2x + 10\).
02

Combine Like Terms

On the left side of the equation, -6 and 7 can be added to get 1. This gives: \(3x + 1 = 2x + 10\).
03

Move Variable Terms to One Side

Subtract 2x from both sides of the equation to isolate the variable amount on one side of the equation. This gives: \(x + 1 = 10\).
04

Solve For the Variable x

Subtract 1 from both sides of the equation to solve for x. This results in: \(x = 9\).

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Key Concepts

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

Distribute Multiplication
When solving linear equations, a key concept is distributing multiplication over addition or subtraction within parentheses. This means applying the multiplication operation to each term inside the parentheses. Let's look at an example. In the equation:\[3(x - 2) + 7 = 2(x + 5)\]You need to distribute the coefficient 3 across the terms within the parentheses. Multiply 3 by both \(x\) and \(-2\):
  • \(3 \times x = 3x\)
  • \(3 \times (-2) = -6\)
So, \(3(x - 2)\) becomes \(3x - 6\). On the right side, similarly distribute the 2:
  • \(2 \times x = 2x\)
  • \(2 \times 5 = 10\)
This turns \(2(x + 5)\) into \(2x + 10\). Distributing multiplication like this helps us simplify the equation, making it easier to solve.
Combine Like Terms
After distributing, the next step is to combine like terms. Like terms are terms that contain the same variable raised to the same power or constants. In our example, after distribution, we have the equation:\[3x - 6 + 7 = 2x + 10\]Look for terms on each side of the equation that can be combined. On the left side, you can combine the constant terms \(-6\) and \(+7\):
  • \(-6 + 7 = 1\)
Thus, the equation simplifies to:\[3x + 1 = 2x + 10\]Combining like terms helps in reducing the number of terms, simplifying the equation and bringing us closer to isolating the variable. By keeping the terms tidy, we're able to handle the operations more smoothly.
Solve for the Variable
Now that we have combined like terms, the equation becomes easier to manage. The next goal is to isolate the variable, which means getting the variable alone on one side of the equation. We start with:\[3x + 1 = 2x + 10\]First, we need to move the variable terms to one side. Subtract \(2x\) from both sides:
  • \(3x - 2x = x\)
This gives:\[x + 1 = 10\]Next, to solve for \(x\), get rid of the constant term on the left side by subtracting 1 from both sides:
  • \(x + 1 - 1 = x\)
  • \(10 - 1 = 9\)
Now, you have:\[x = 9\]By isolating the variable, we determine its value, which solves the equation. This step-by-step approach helps in clearly understanding how each operation affects the equation.

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Most popular questions from this chapter

In Exercises \(112-123,\) use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A truck can be rented from Basic Rental for \(\$ 50\) per day plus \(\$ 0.20\) per mile. Continental charges \(\$ 20\) per day plus \(\$ 0.50\) per mile to rent the same truck. How many miles must be driven in a day to make the rental cost for Basic Rental a better deal than Continental's?

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