Problem 15 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. \(6 x-(3 x+10)=14\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 8\).

Step 1: Simplify the Equation

First, distribute the minus sign across the terms inside the bracket: \(6x - 3x - 10 = 14\)

Step 2: Group Similar Terms

Combine like terms on the left side of the equation: \(3x - 10 = 14\)

Step 3: Isolate the Variable

Isolate the variable by adding 10 to both sides of the equation: \(3x = 24\)

Step 4: Solve for the Variable

Finally, solve for \(x\) by dividing both sides of the equation by 3: \(x = 8\)

Step 5: Verify the Solution

Verify the solution by substituting \(x = 8\) back into the original equation: \(6(8) - (3(8) + 10) = 48 - (24 + 10) = 48 - 34 = 14\), which is the right hand side of the original equation. Thus, \(x = 8\) is the correct solution.

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is a foundational skill in algebra. An algebraic expression is a combination of numbers, variables (like x), and operators (such as addition, subtraction, multiplication, and division) that represents a particular quantity.

In the given exercise, the equation begins with an algebraic expression, 6x - (3x + 10). To simplify it, we distribute the minus sign to the terms in the brackets. This process involves multiplying the negative one (-1) from the subtraction operation outside the bracket by each term inside the bracket. It simplifies to 6x - 3x - 10, combining like terms to reduce it further. Like terms are terms that have the same variables raised to the same power. Here, 6x and -3x are like terms, and combining them simplifies the expression to 3x - 10.

Isolating Variables

Isolating the variable is a critical step in solving linear equations; it means rearranging the equation so that the variable you are solving for is alone on one side of the equation. The goal is to have the variable by itself, with a coefficient of one.

The process includes performing inverse operations to move terms from one side of the equation to the other until the variable is isolated. In the example, after simplifying the expression, we aimed to isolate x by adding 10 to both sides, resulting in 3x = 24. From there, we divided both sides by the coefficient of x, which is 3, to get x = 8. This step is vital because it provides an apparent value for the variable.

Verifying Solutions

After solving for the variable, it is important not just to assume the solution is correct, but to verify it. This verification is done by substituting the solution back into the original equation and checking if the left side equals the right side.

In our exercise, after deducing that x = 8, we verified the solution by replacing each instance of x in the original equation with 8 and simplifying. Once we calculated and ended up with the same value on both sides of the original equation, we confirmed that the solution is indeed correct. This step not only validates your results but also helps to catch any potential arithmetic errors made along the way.