Problem 32 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(-6 x+2 x-11=-2(2 x-3)+4\)

Step-by-Step Solution

Answer

The equation is a contradiction; there is no solution.

Step 1: Simplify both sides of the equation

Simplify the terms on each side of the equation. Start with the left side: ewline -6x + 2x - 11 = -4x - 11ewline Then simplify the right side: ewline -2 (2x - 3) + 4 = -4x + 6 + 4 = -4x + 10

Step 2: Combine like terms

Combine the like terms on both sides to get a simpler equation: ewline -4x - 11 = -4x + 10

Step 3: Isolate terms with variable

Add 4x to both sides to isolate the constants: ewline -4x + 4x - 11 = -4x + 4x + 10 ewline -11 = 10

Step 4: Check for Identity or Contradiction

Since -11 does not equal 10, this equation is a contradiction. There are no values of x that will satisfy this equation.

Key Concepts

Simplify Equations

Simplifying equations is the first and a critical step when you are solving algebraic problems. You aim to reduce each side of the equation to its simplest form. This makes it easier to manage and understand. In our example, you start with -6x + 2x - 11 = -2(2x - 3) + 4.
To simplify, handle each side independently:

• On the left side, combine -6x and 2x to get -4x - 11.
• For the right side, apply the distributive property to deal with the parentheses: -2(2x - 3) becomes -4x + 6. Then add 4 to get -4x + 10.

Now, the equation looks simpler: -4x - 11 = -4x + 10. This step sets the stage for combining like terms and identifying contradictions.

Combine Like Terms

Combining like terms helps in creating a simplified version of the equation. This involves merging coefficients of the same variables and dealing with constants separately. In our equation, after simplifying, we have -4x - 11 = -4x + 10.
Check both sides of the equation to see if there are any terms you can combine:

• Both sides have -4x, which can be simplified further.

When you add 4x to both sides, you essentially eliminate the variable term, making the constants stand out:
-4x + 4x - 11 = -4x + 4x + 10. This simplifies to -11 = 10, shedding light that only constants are left.

Identify Contradictions in Equations

The final step is to check for contradictions. Contradictions occur when no value can satisfy the equation.
After combining like terms in our equation, we are left with -11 = 10. This statement is false because -11 is not equal to 10, leading to a contradiction.

• If an equation simplifies to a false statement, it means there are no values of the variable that can make the equation true.
• When such contradictions appear, it signals that the equation has no solution.

In this case, -6x + 2x - 11 = -2(2x - 3) + 4 is a contradiction since both sides simplify to a false statement. Recognizing these contradictions saves time and effort in trying to find solutions where none exist.