Problem 31 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(-2 x+5 x-9=3(x-4)-5\)

Step-by-Step Solution

Answer

No solution; the equation is a contradiction.

Step 1 - Combine like terms on the left side

Combine the like terms on the left side of the equation: -2x + 5x - 9 = 3(x - 4) - 5This simplifies to: 3x - 9 = 3(x - 4) - 5

Step 2 - Distribute on the right side

Distribute the 3 on the right side of the equation: 3x - 9 = 3x - 12 - 5

Step 3 - Simplify the right side

Combine the constants on the right side: 3x - 9 = 3x - 17

Step 4 - Move variable terms to one side

Subtract 3x from both sides of the equation: 3x - 9 - 3x = 3x - 17 - 3xThis simplifies to: -9 = -17

Step 5 - Determine if the statement is true

The equation -9 = -17is a false statement, indicating no solution.

Step 6: Conclusion

Since the equation resulted in a false statement, it means that there is no solution to the equation. Thus, it is a contradiction.

Key Concepts

Combining Like Terms

When solving linear equations, you often encounter like terms, which are terms that have the same variable raised to the same power. Combining like terms simplifies the equation and makes it easier to solve. For instance, in the equation -2x + 5x - 9 = 3(x - 4) - 5the terms -2x and 5x both have the variable 'x'. You can combine them easily:-2x + 5x = 3x So, the equation becomes: 3x - 9 = 3(x - 4) - 5Always remember to keep the variable and its coefficient together when combining like terms.

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions involving parentheses. It states that a(b + c) = ab + acIn our equation 3(x - 4) - 5we apply the distributive property to get rid of the parentheses: 3 * x - 3 * 4 - 5which simplifies to: 3x - 12 - 5. Applying the distributive property early in an equation allows you to simplify and combine like terms more effectively later.

No Solution Equations

Sometimes when you solve linear equations, you might find that no solution exists. This happens when you end up with a false statement after simplifying the equation. For example, after simplifying and combining like terms in 3x - 9 = 3x - 17we subtract 3x from both sides, resulting in:-9 = -17Since -9 does not equal -17, the equation has no solution. This means there are no values of 'x' that would make the original equation true.

Contradiction in Equations

A contradiction in an equation occurs when you arrive at a statement that is mathematically impossible, indicating that no value of the variable could satisfy the original equation. For instance, in -9 = -17you end up with an untrue statement after solving the equation. Since it's impossible for -9to equal -17the original equation is considered a contradiction. Recognizing contradictions is essential in determining whether or not an equation has solutions.