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Chapter 2: Problem 33

Solve each equation, and check your solution. $$ 6(4 x-1)=12(2 x+3) $$

Short Answer

Expert verified
The equation has no solution because it simplifies to a contradiction.

Step by step solution

01

Distribute on both sides

Distribute the constants outside the parentheses on both sides of the equation. On the left side: \(6 \times (4x) - 6 \times 1\). On the right side: \(12 \times (2x) + 12 \times 3\). This simplifies to: \[24x - 6 = 24x + 36\]
02

Simplify the equation

Simplify the equation by moving all the terms involving \(x\) to one side and constant terms to the other side. Here, subtract \(24x\) from both sides: \[24x - 6 - 24x = 24x + 36 - 24x\] This reduces to: \[-6 = 36\]
03

Analyze the result

Since the equation \(-6 = 36\) is a contradiction and does not hold true, it implies that there's no solution for this equation. Therefore, the equation has no solution.
04

Verification

Since the simplified equation is a contradiction, no further verification is needed. This confirms that there's no solution for the original equation.

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Key Concepts

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

distributive property
The distributive property is a fundamental concept in algebra. It shows how to distribute multiplication over addition or subtraction. For example, in the equation \(6(4x - 1)\), you distribute the 6 to both terms inside the parentheses:
\(6 \times 4x - 6 \times 1\).
The same goes for the right side of the equation: \(12(2x + 3)\) becomes \(12 \times 2x + 12 \times 3\).
This property helps in breaking down complex expressions into simpler parts.

Understanding the distributive property is crucial as it is widely used in algebra to simplify and solve equations.
simplifying equations
Simplifying equations involves combining like terms and arranging them in a more manageable form. After distributing on both sides, our equation becomes:
\[ 24x - 6 = 24x + 36 \]
Notice that we have \(24x\) on both sides. To simplify, we subtract \(24x\) from both sides:
\[ 24x - 6 - 24x = 24x + 36 - 24x \]
This reduces to:
\[ -6 = 36 \]
Simplifying equations is a key step in solving them. Always ensure like terms are on the same side, which makes it easier to isolate the variable.
contradiction in equations
A contradiction in equations occurs when you end up with a statement that is always false. For instance, after simplifying our equation, we get:\(-6 = 36\).
This is clearly false, as \(-6\) can never equal \(36\).
Such contradictions indicate that there is no possible value for \(x\) that satisfies the original equation.
Contradictions are important because they tell us that the equation has no solution.
no solution in equations
When we find a contradiction while solving an equation, it means there is no solution. In our example, the final simplified equation was \(-6 = 36\).
Since this is never true, we conclude that the equation \(6(4x - 1) = 12(2x + 3)\) has no solution.

Recognizing when an equation has no solution is important. It prevents us from wasting time looking for an answer that doesn't exist.
Always check for contradictions to determine if an equation has no solution.

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