Problem 55 Solve and check. Label any contr... [FREE SOLUTION]

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Chapter 2: Problem 55

Solve and check. Label any contradictions or identities. $$4+7 a=7(a-1)$$

Short Answer

Expert verified

The equation has no solution due to a contradiction (4 鈮� -7).

Step by step solution

Expand the Right Side

Expand the expression on the right side of the equation: \[4 + 7a = 7(a - 1)\] becomes: \[4 + 7a = 7a - 7\]

Eliminate the Variable Term

Subtract $7a$ from both sides to eliminate the variable term on one side: \[4 + 7a - 7a = 7a - 7 - 7a\] This simplifies to: \[4 = -7\]

Identify Contradiction

Observe that $4 = -7$ is a contradiction because it is not possible for these two values to be equal. Since we reached a contradiction, it means there is no solution to the equation.

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Key Concepts

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

Algebraic Equations

Algebraic equations are mathematical statements that prove the equality between two expressions. In the equation given ($4 + 7a = 7(a-1)$), we have an equation involving the variable $a$. Understanding the components of algebraic equations is essential:

**Terms**: individual parts of the equation, separated by a plus (+) or minus (-) sign.
**Coefficients**: numerical factors that multiply the variables, like 7 in the term 7a.
**Constants**: numbers without variables, like 4 in this equation.

Using these components correctly helps in manipulating and solving equations effectively. Always keep track of these parts while simplifying or expanding equations.

Contradictions in Equations

Contradictions arise in equations when you end up with an impossible statement. In our example, by simplifying both sides we ended up with $4 = -7$. This is clearly untrue, which reveals a contradiction. Because of this contradiction, we can conclude there is no solution to the equation. Here are some steps to identify contradictions:

**Simplify both sides of the equation fully**: This might involve expanding expressions, combining like terms, or isolating variables.
**Compare final expressions**: Check if the simplified expressions make sense; if they don鈥檛 (like $4 = -7$), then it's a contradiction and the equation has no solution.

This process will help you confidently identify when and why no solution exists.

Equation Simplification

Simplification is key to solving equations. It involves reducing complex equations to simpler, more manageable forms. In the given example, simplification steps included:

**Expanding the right side**: $7(a - 1)$ was expanded to $7a - 7$, making both sides comparable.
**Eliminating terms**: Subtracting $7a$ from both sides to isolate the remaining terms.
**Simplifying constants**: Combining or clearing constants to see if the equation holds true.

By consistently applying these rules, you can simplify most algebraic equations. It鈥檚 important to perform the same operation on both sides of the equation to maintain equality, and always check your steps for accuracy. This methodical approach will help solve or identify issues in equations efficiently.

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91影视

Solve and check. Label any contradictions or identities. $$4+7 a=7(a-1)$$

Short Answer

Step by step solution

Expand the Right Side

Eliminate the Variable Term

Identify Contradiction

Key Concepts

Algebraic Equations

Contradictions in Equations

Equation Simplification

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