Problem 52 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using both the addition and multiplication properties of equality. Check proposed solutions. $$9 x+2=6 x-4$$

Step-by-Step Solution

Answer

The solution to the equation \(9x + 2 = 6x - 4\) is \(x = -2\).

Step 1: Simplify the Equation

Start with the given equation, \(9x + 2 = 6x - 4\). To get all terms with 'x' on the same side, subtract '6x' from each side, giving you \(9x - 6x + 2 = 6x - 6x - 4\), which simplifies to \(3x + 2 = -4\).

Step 2: Solve for 'x'

Next, subtract '2' from each side to isolate the 'x' variable: \(3x + 2 - 2 = -4 - 2\), which results in \(3x = -6\). To fully isolate 'x', divide each side by '3': \(\frac{3x}{3} = \frac{-6}{3}\), which leads to the solution 'x = -2'.

Step 3: Check the Solution

Substitute '-2' in place of 'x' in the initial equation to verify. You get: \(9(-2) + 2 = 6(-2) - 4\), which simplifies to \(-16 = -16\). Both sides are equal, thus the proposed solution is correct.

Key Concepts

Addition Property of Equality

The addition property of equality is a crucial concept in solving equations. It states that if you add the same number to both sides of an equation, the equation remains balanced. This means that the equality holds true, and you haven't introduced any errors.
To apply this principle in the equation solving process, you can use it to simplify equations and make them easier to solve. In our example, we used the addition property to address the equation:

• The original equation was: \(9x + 2 = 6x - 4\).
• We subtracted \(6x\) from both sides to bring all \(x\) terms to one side, making it \(9x - 6x + 2 = -4\).

This action ensured the equation stays balanced, allowing for simpler calculations.

Multiplication Property of Equality

The multiplication property of equality is another vital tool. It states that you can multiply or divide both sides of an equation by the same nonzero number without disrupting the balance of the equation. This property is especially useful for isolating variables.
In solving our example, we reached the equation \(3x = -6\) after applying the addition property.
To solve for \(x\), we used the multiplication property:

• We divided both sides by \(3\), i.e., \(\frac{3x}{3} = \frac{-6}{3}\).
• This simplified our equation to \(x = -2\).

By dividing both sides by the coefficient of \(x\), we successfully isolated \(x\) to find its value.

Checking Solutions

Checking solutions is an important final step to ensure our solution is correct. After solving an equation, it's always good practice to substitute the found values back into the original equation to verify the solution.
In our example, we determined that \(x = -2\). By substituting \(-2\) back into the original equation \(9x + 2 = 6x - 4\), we check for consistency:

• Substitute: \(9(-2) + 2 = 6(-2) - 4\).
• Simplify: \(-18 + 2 = -12 - 4\).
• Result: \(-16 = -16\).

Since both sides of the equation are equal, our solution is verified as correct. Checking helps in confirming that no error was made during calculations.