Problem 63 - Introductory Algebra for College Students

Question

Solve each equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. $$7+2(3 x-5)=8-3(2 x+1)$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{2}{3}\).

Step 1: Apply Distributive Property

Distribute the values outside the parentheses to all the values within. So, \(7 + 2(3x - 5) = 8 - 3(2x + 1)\) simplifies into: \(7 + 6x - 10 = 8 - 6x - 3\).

Step 2: Simplify

Combine like terms on the left-hand side (LHS) and right-hand side (RHS) of the equation. This gives: \( -3 + 6x = 5 - 6x \).

Step 3: Isolate the Variable x

Moving \(6x\) from LHS to RHS gives: \( -3 = 5 - 12x \). Further simplifying by moving constants out and reversing the equation, we get: \(12x = 5 + 3 \).

Step 4: Solving for x

Final step is to divide by \(12\) on both sides to isolate \(x\), so we get: \(x = \frac{8}{12} = \frac{2}{3}\).

Key Concepts

Distributive Property

Understanding the distributive property is key to solving equations when there are parentheses involved. This mathematical property allows you to multiply a single term across terms inside parentheses. In the original exercise, we have terms like \(2(3x - 5)\) and \(-3(2x + 1)\).

• For the expression \(2(3x - 5)\), you'll distribute the \(2\) to both \(3x\) and \(-5\), resulting in \(6x - 10\).
• Similarly, \(-3(2x + 1)\) becomes \(-6x - 3\) after applying the distributive property.

Breaking down the expression step by step helps in grasping the changes each term undergoes. By understanding this, we make sure that every part of the expression is accurately expanded, maintaining the equation's integrity.

Isolate Variable

Once you've expanded and simplified both sides of the equation, the next step is to isolate the variable. Isolating the variable means rearranging the equation so the variable, typically \(x\), is one side alone.
This is done by moving all terms involving \(x\) to one side and constants to the other.
In the solution of our exercise, you saw that after simplifying, we have \(-3 + 6x = 5 - 6x\). From here, we moved \(6x\) from the left side to the right side, which requires adding \(6x\) to both sides, giving \( -3 = 5 - 12x\).
This type of manipulation ensures that you can focus solely on solving for \(x\) without extra numbers crowding the equation.

Combine Like Terms

Combining like terms is a crucial part of simplifying equations. This involves adding or subtracting terms that have identical variables raised to the same power, or simply merging constant terms.
In the example given, after distributing the terms, we noticed expressions that could be combined:

• On the left side, terms \(7\) and \(-10\) can be combined to give \(-3\).
• Similarly, on the right side, \(8\) and \(-3\) give \(5\).

Combining like terms simplifies the equation and makes the next steps more straightforward. It reduces messy, long equations into more manageable pieces, which is especially helpful when isolating variables.

Simplifying Equations

Simplifying equations is essentially tidying things up. It involves using the properties we discussed — like distribution and combining like terms — to make the equation as straightforward as possible.
You need to ensure each side of your equation is reduced to its simplest form. In the exercise, simplifying \(7 + 6x - 10 = 8 - 6x - 3\) results in \(-3 + 6x = 5 - 6x\).
Simplification highlights the relationships between numbers and terms, allowing you to isolate variables more easily.
Think of it as decluttering the equation. You want to see the key components — the terms involving \(x\) and the constants — clearly, which leads to quicker and more efficient problem-solving.