Problem 39 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(6 x-4(3-2 x)=5(x-4)-10\)

Step-by-Step Solution

Answer

The solution is \(x = -2\). The equation is an identity.

Step 1 - Distribute

Distribute the constants inside the parentheses: \(6x - 4(3 - 2x) = 5(x - 4) - 10\). This yields: \(6x - 12 + 8x = 5x - 20 - 10\).

Step 2 - Combine Like Terms

Combine like terms on both sides of the equation: \(6x + 8x - 12 = 5x - 30\). Simplify to get: \(14x - 12 = 5x - 30\).

Step 3 - Isolate Variable on One Side

Move all terms involving \(x\) to one side and constants to the other side. Subtract \(5x\) from both sides: \(14x - 5x - 12 = -30\). This simplifies to: \(9x - 12 = -30\).

Step 4 - Solve for \(x\)

Add 12 to both sides of the equation to isolate the term with \(x\): \(9x - 12 + 12 = -30 + 12\). This simplifies to: \(9x = -18\). Now, divide both sides by 9: \(x = -2\).

Step 5 - Check the Solution

Substitute \(x = -2\) back into the original equation to verify the solution. The original equation is: \(6x - 4(3 - 2x) = 5(x - 4) - 10\). Substitute \(x = -2\): \(6(-2) - 4(3 - 2(-2)) = 5(-2 - 4) - 10\). Simplify both sides: \(-12 - 4(3 + 4) = -30 - 10\). This further simplifies to: \(-12 - 28 = -40\) and \(-40 = -40\) which confirms the solution is correct.

Key Concepts

Distributing Constants

When solving linear equations, you often need to distribute constants. This means you multiply the constant by each term inside the parentheses. For example, in the equation 6x - 4(3 - 2x) = 5(x - 4) - 10, we start by distributing the constants inside the parentheses.
When we distribute -4 through (3 - 2x), we get:
-4 * 3 and -4 * -2x, resulting in -12 and +8x. This changes the equation to: 6x - 12 + 8x.
Similarly, distribute 5 through (x - 4) to get 5 * x and 5 * - 4, resulting in 5x and -20.
After distributing, the equation becomes: 6x - 12 + 8x = 5x - 20 - 10.

Combining Like Terms

After distributing, identify and combine like terms to simplify the equation further. Like terms are terms that have the same variable raised to the same power. In our same example: It changes from: 6x - 12 + 8x = 5x - 20 - 10
To: 14x - 12 = 5x - 30, by combining 6x and 8x to get 14x.
Similarly, combining -20 and -10 on the right side gives us -30,
Measurement combining helps in reducing the number of terms in the equation, making it easier to solve.

Isolating Variables

To solve for a variable, you need to isolate it on one side of the equation. This involves moving terms with the variable you want to isolate to one side and constants to the other. In our example: 14x - 12 = 5x - 30
Move all terms involving x to one side by subtracting 5x from both sides: 14x - 5x - 12 = -30, giving us: 9x - 12 = -30.
Next, move the constant term (in this case, -12) to the other side by adding 12 to both sides: 9x - 12 + 12 = -30 + 12, resulting in: 9x = -18.
Lastly, isolate x by dividing both sides by 9: x = -2. By isolating the variable, you can solve for the exact value.