Problem 52 - Introductory Algebra for College Students

Question

Solve each equation in 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

Begin by moving variables with the term \(x\) to one side of the equation, and constants to the other. Subtract \(6x\) from both sides of the equation to move the \(x\) terms to the left, yielding \(3x + 2 = -4\).

Step 2: Isolate \(x\)

Next, subtract \(2\) from both sides of the equation in order to isolate \(x\) by itself on the left side. This will yield \(3x = -6\).

Step 3: Solve for \(x\)

Finally, divide by \(3\) on both sides of the equation to solve for \(x\). This will give \(x = -2\).

Step 4: Check the proposed solution

Substitute \(-2\) for \(x\) in the original equation \(9x + 2 = 6x - 4\). This gives \(9(-2) + 2 = 6(-2) - 4\), which simplifies to \(-18 + 2 = -12 - 4\) and further simplifies to \(-16 = -16\), which is true. Therefore, the solution for \(x\) is \(-2\).

Key Concepts

Addition Property of Equality

The addition property of equality is a fundamental concept in algebra that helps maintain the balance of an equation. It states that if you add the same value to both sides of an equation, the equation remains true. This concept can be imagined like a balanced scale. If you add the same weight to both sides, the scale remains balanced.
In the given problem, the addition property of equality was used in a crucial step. Here’s how:

• Initially, we had the equation with the variable terms and constant terms mixed: \( 9x + 2 = 6x - 4 \).
• To separate the \( x \) terms, we subtracted \( 6x \) from both sides. This ensures that the equation stays balanced while moving like terms together.
• The equation becomes \( 3x + 2 = -4 \).

By applying the addition property of equality in these steps, we've made it easier to isolate the variable and finally solve the equation.

Multiplication Property of Equality

Once you have simplified an equation using addition or subtraction, the multiplication property of equality helps you solve for the variable by getting rid of any coefficients. This property states that multiplying both sides of an equation by the same nonzero number keeps the equation balanced.
In this exercise, after isolating the variable term \( 3x = -6 \), we use multiplication (or division, which is effectively multiplication by the reciprocal) to solve for \( x \):

• Since \( 3x \) is the result of our previous steps, we divide each side by 3 to relate \( x \) alone: \( \frac{3x}{3} = \frac{-6}{3} \).
• This simplifies the equation to \( x = -2 \).

By applying the multiplication property of equality, we effectively find the solution to the equation, where \( x \) equals -2. This demonstrates how powerful this property can be in solving linear equations.

Solving Linear Equations

Solving linear equations involves a series of strategic steps that use properties of equality to find the value of the variable. A linear equation is an equation that, when graphed, produces a straight line. Understanding how to solve them is key to mastering algebra.
Here is a breakdown of how you can solve any linear equation such as \( 9x + 2 = 6x - 4 \):

• Step 1: Rearrange the equation. First, you want to collect all variable terms on one side and constant terms on the other. Use the addition or subtraction property of equality here.
• Step 2: Simplify and isolate the variable. Focus on getting the variable on one side of the equation, leaving all constants on the opposite side. Simplification can involve adding or subtracting terms.
• Step 3: Use multiplication or division to solve for the variable. Lastly, if your variable has a coefficient, divide both sides by it to solve for the variable directly.

Each of these steps is a piece of the puzzle to effectively and efficiently solving linear equations, leading you to a solution where the equation holds true.