Problem 38 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(6 x-3(5 x+2)=4(1-x)\)

Step-by-Step Solution

Answer

The solution is \(x = -2\). The equation is neither an identity nor a contradiction.

Step 1: Distribute and Simplify

First, distribute the constants through the parentheses: For the left side: \(6x - 3(5x + 2) = 6x - 15x - 6 = -9x - 6\) For the right side: \(4(1 - x) = 4 - 4x\)

Step 2: Set the Simplified Expressions Equal to Each Other

Now the equation becomes: \(-9x - 6 = 4 - 4x\)

Step 3: Collect Like Terms

Add \(4x\) to both sides to start isolating the variable: \( -9x + 4x - 6 = 4 - 4x + 4x\) Simplify this to get: \(-5x - 6 = 4\)

Step 4: Isolate the Variable

Add 6 to both sides to further isolate \(x\): \( -5x - 6 + 6 = 4 + 6\) Simplify to: \(-5x = 10\)

Step 5: Solve for the Variable

Divide both sides by \(-5\): \(x = \frac{10}{-5} \)Therefore, \(x = -2\)

Step 6: Check the Solution

Substitute \(x = -2\) back into the original equation to verify: \(6(-2) - 3(5(-2) + 2) = 4(1 - (-2))\) Simplify both sides: \(-12 - 3(-10 + 2) = 4(1 + 2)\)\(-12 - 3(-8) = 4(3)\)\(-12 + 24 = 12\)\(12 = 12\)Since both sides are equal, \(x = -2\) is a correct solution.

Key Concepts

Equation Solving Steps

In algebra, solving an equation requires following a series of steps meticulously to find the value of the variables involved. Here's a breakdown of the standard steps:

1. **Distribute and Simplify**: Spread out any constants or variables within parentheses to simplify the expressions on both sides of the equation. For instance, in our example, we distribute constants on the left and right sides:
- Left side: \[ 6x - 3(5x + 2) = 6x - 15x - 6 = -9x - 6 \] - Right side: \[ 4(1 - x) = 4 - 4x \]
2. **Set Equal**: Make sure that the simplified expressions on both sides of the equation are set equal to each other. This looks like:
\[ -9x - 6 = 4 - 4x \]
3. **Collect Like Terms**: Group and combine similar terms to simplify the equation further. Add or subtract terms as needed to isolate the variable term on one side of the equation.
\[ -9x + 4x - 6 = 4 - 4x + 4x \] simplifies to \[ -5x - 6 = 4 \]
4. **Isolate the Variable**: Continuously isolate the variable by performing arithmetic operations on both sides. For instance, adding 6 to both sides:
\[ -5x - 6 + 6 = 4 + 6 \] which simplifies to \[ -5x = 10 \]
5. **Solve for the Variable**: Finally, solve for the variable by performing the necessary division or multiplication. Divide both sides by -5 to find:
\[ x = \frac{10}{-5} \] resulting in \[ x = -2 \]By following these steps, you ensure you’ve thoroughly and correctly solved the equation.

Checking Solutions

To ensure your solution is correct, it's essential to check it by substituting the value back into the original equation. This validation process confirms if the left side equals the right side:

For our example, substitute \( x = -2 \) back into the original equation:
\[ 6(-2) - 3(5(-2) + 2) = 4(1 - (-2)) \]
Simplify both sides:
- Left: \[ -12 - 3(-10 + 2) = -12 - 3(-8) = -12 + 24 = 12 \] - Right: \[ 4(1 + 2) = 4(3) = 12 \]Since both sides are equal, \( x = -2 \) is indeed a correct solution.

Checking solutions ensures accuracy. It helps identify any arithmetic errors made while solving.

Identity versus Contradiction

In algebra, equations can sometimes be classified as identities or contradictions based on the nature of their solutions. Understanding these terms improves problem-solving skills.

- **Identity**: An equation that is true for all possible values of the variables. For instance,
\[ 2(x + 3) = 2x + 6 \] is always true, regardless of the value of \ x \. Any value will satisfy this equation.

- **Contradiction**: An equation that has no solution because it is never true. An example is:
\[ x + 3 = x - 4 \]. After simplifying, you get:
\[ 3 = -4 \], which is invalid.

In our original exercise, we found a specific solution (\( x = -2 \)), confirming that the equation is neither an identity nor a contradiction, but instead, it has a unique solution.

Classifying equations helps make sense of their nature and the possible solutions they might have.