Problem 56 - Algebra A Combined Function

Question

Solve. See Examples 1 through 7 $$ 4(2+x)+1=7 x-3(x-2) $$

Step-by-Step Solution

Answer

The equation has no solution.

Step 1: Distribute Through Parentheses

Start by distributing the coefficients across the terms inside the parentheses on both sides of the equation. For the left side: \(4(2+x)\) becomes \(8 + 4x\). For the term \(-3(x-2)\) on the right side, distribute to get \(-3x + 6\).

Step 2: Simplify Each Side of the Equation

After distributing, the equation becomes:\[8 + 4x + 1 = 7x - 3x + 6\]Simplify each side: the left side simplifies to \(9 + 4x\) and the right side simplifies to \(4x + 6\).

Step 3: Move Variable Terms to One Side

Subtract \(4x\) from both sides to move the variable terms from one side of the equation:\[9 = 4x + 6 - 4x\]This simplifies to \(9 = 6\).

Step 4: Move Constant Terms to One Side

Since \(4x\) has been canceled out, subtract \(6\) from both sides to isolate the constant terms:\[9 - 6 = 6 - 6\]This simplifies to \(3 = 0\), which is a contradiction.

Step 5: Interpret the Result

The equation \(3 = 0\) suggests there is no solution. The original equation has no value of \(x\) that would satisfy the equation.

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra that helps in simplifying expressions. It allows you to expand expressions by multiplying a single term by each term within a set of parentheses. This is crucial in breaking down complex problems. Imagine you have the expression \(a(b+c)\). By applying the distributive property, you multiply \(a\) with each term inside the parentheses: \(ab + ac\).
In our exercise, we apply the distributive property to both sides of the equation. On the left side, the expression \(4(2+x)\) expands to \(8 + 4x\). On the right side, \(-3(x-2)\) expands to \(-3x + 6\). Using this property makes the equation more straightforward to solve or rearrange for further simplification.

Simplifying Equations

Simplifying equations involves reducing them to their simplest form while maintaining equality. After distributing terms, the next step is usually combining like terms. Like terms have the same variable parts raised to the same power and can be combined by adding or subtracting their coefficients.
In our case, after distributing, we had the equation \(8 + 4x + 1 = 7x - 3x + 6\). Here, simplifying involves combining \(8\) and \(1\) on the left, resulting in \(9 + 4x\), and \(7x - 3x\) on the right, which simplifies to \(4x + 6\). This reduction makes it easier to balance the equation by isolating variables and constants on one side each.
Simplification makes complex algebraic problems more intuitive and manageable to solve.

No Solution in Algebra

In algebra, sometimes you may end up with an equation that has no solution. This happens when all possible attempts to isolate the variable lead to a contradiction—an untrue statement.
For instance, in our problem, after simplifying and rearranging, we ended up with \(9 = 6\) or eventually \(3 = 0\). This result is a contradiction because \(3\) will never equal \(0\).
When encountering such outputs, it indicates that no value for the variable can satisfy the equation. It means there is no solution or the solution set is empty. Recognizing scenarios of 'no solution' is critical because it helps identify and discard erroneous assumptions early in problem-solving. Understanding why a problem has no solution can also lead to deeper insights into the problem's structure.