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Chapter 2: Problem 106

Solve each equation, if possible. $$ 5(x+3)-3 x=2(x+8) $$

Short Answer

Expert verified
The equation has no solution; it results in a contradiction (15 鈮� 16).

Step by step solution

01

Expand the Equation

First, expand both sides of the given equation by distributing the constants 5 and 2 across the terms inside the parentheses.\[5(x+3) - 3x = 2(x+8)\]This results in:\[5x + 15 - 3x = 2x + 16\]
02

Combine Like Terms

Next, simplify both sides of the equation by combining like terms. Combine the terms with \(x\) on the left.\[(5x - 3x) + 15 = 2x + 16\]This simplifies to:\[2x + 15 = 2x + 16\]
03

Isolate Variable Terms

Attempt to isolate \(x\) by getting all terms involving \(x\) on one side and constants on the other. Here, subtract \(2x\) from both sides.\[2x + 15 - 2x = 2x + 16 - 2x\]Results in:\[15 = 16\]
04

Analyze the Result

Notice the equation \(15 = 16\). This is a contradiction, meaning there is no value of \(x\) that can satisfy the equation. Therefore, the original equation has no solution.

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Key Concepts

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

Solving Equations
Solving equations is a fundamental aspect of algebra. It involves finding the value of a variable that makes an equation true. The general goal is to isolate the variable on one side of the equation. This means arranging the equation so that the variable is by itself on one side, with all the constant terms on the other side. Simple steps to solve equations include:
  • Expanding any terms using distribution.
  • Combining like terms to simplify the equation.
  • Isolating the variable by performing operations on both sides.
  • Checking for any contradictions or special cases that imply no solution or infinitely many solutions.
In our problem, when we tried to solve the equation, we ended up with an impossible statement (15 = 16). Such contradictions indicate that the equation has no solution.
Distributive Property
The distributive property is a crucial algebraic rule used to simplify equations. It allows us to "distribute" a number outside a parenthesis to each term inside the parenthesis. For a number 5 distributed over the terms in the parenthesis, it looks like this: \[ a(b + c) = ab + ac \]
In our original equation, we applied this property twice:
  • Distribute the 5 over \(x+3\): \(5 imes (x+3) = 5x + 15\).
  • Distribute the 2 over \(x+8\): \(2 imes (x+8) = 2x + 16\).
This step is essential as it transforms the equation into a simpler form with no parenthesis. Proper use of the distributive property helps in combining like terms and further simplifies problem-solving.
Like Terms
Combining like terms is a fundamental step in simplifying equations. Like terms are terms that have the same variable raised to the same power. You can add or subtract these terms with ease. When simplifying, remember:
  • Collect all the terms that have the variable on the same side.
  • Combine them into a single term.
For example, in our equation:\[ 5x - 3x + 15 = 2x + 16 \]
Here, 5x and -3x are like terms because they both have the variable \( x \). By combining them, we simplify the left side to 2x. This consolidation reduces clutter in the equation, making the next steps clearer and manageable. Always look for and combine like terms to support accurate problem-solving throughout your algebraic journey.

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Most popular questions from this chapter

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