Problem 63 - College Algebra

Question

The equations in Exercises \(59-70\) combine the types of equations we have discussed in this section. Solve each equation or state that it is true for all real numbers or no real numbers. $$ 8 x-(3 x+2)+10=3 x $$

Step-by-Step Solution

Answer

The solution is \(x = -4\).

Step 1: Distribute the Subtraction

Distribute the subtraction operation over the terms inside the parentheses: \(8x - 3x - 2 + 10 = 3x\) .

Step 2: Combine like Terms

Combine like terms on both sides of the equation: \(5x + 8 = 3x\).

Step 3: Rearrange the Equation

Rearrange the equation to isolate \(x\) on one side. Subtract \(3x\) from each side of the equation to eliminate \(3x\) on the right side and simplify the left side: \(2x + 8 = 0 \). Then, subtract 8 from each side of the equation: \(2x = -8\).

Step 4: Solve for \(x\)

Finally, solve for \(x\) by dividing each side of the equation by 2: \(x = -4\) .

Key Concepts

Distributive Property

The distributive property is a handy tool when working with equations, especially when you have parentheses involved. It allows you to multiply a single term by each term within the parentheses. In our specific problem, we use distribution to deal with subtraction by considering it as multiplying by -1. This approach helps in simplifying the initial equation, like converting:

• \(8x - (3x + 2) + 10\) becomes \(8x - 3x - 2 + 10\).

This makes it simpler to handle as all terms become separately visible. Applying the distributive property is the initial step that often sets the stage for solving equations by simplifying them deeply.

Combining Like Terms

Once you use the distributive property, the next step is combining like terms. Like terms are those that have the same variable and exponent, or no variable at all. For instance, in our problem, terms like \(8x\) and \(-3x\) are considered like terms because they both contain the variable \(x\). Right after distribution, what you do is:

• Add or subtract these similar terms together.

In the equation \(8x - 3x - 2 + 10\), combining like terms gives: \(5x + 8\). This step is crucial as it simplifies equations further by reducing the number of terms. It gets you closer to isolating the variable you are solving for.

Isolating Variables

After combining like terms, the aim is to isolate the variable, so you can find its value. Isolating a variable means getting it by itself on one side of the equation. In our problem, the initial rearrangement \(5x + 8 = 3x\) shows this stage. You need to clear out the \(3x\) from one side to give us something more straightforward:

• Subtract \(3x\) from both sides which leaves \(2x + 8=0\).

At this point, the positioning is ready, only requiring you to get rid of constant terms. The process of isolating involves strategically using addition, subtraction, multiplication, or division to bring the variable to one side of the equation.

Equation Rearrangement

Rearranging equations is about moving terms around to solve them. Once like terms are combined and the variable is isolated, you have to finalize the rearrangements to solve for it. In our solution, when \(2x + 8 = 0\) is achieved, subsequent steps involve:

• Subtract 8 to remove it from one side, leading to \(2x = -8\).
• Finally, divide both sides by 2 to completely isolate and determine \(x\), giving \(x = -4\).

This final rearrangement is crucial in confirming and solving for the variable precisely. Handling equations with ease often means being skillful in juggling numbers and terms around, ensuring the balance of an equation while achieving your required outcome.