Problem 51 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using both the addition and multiplication properties of equality. Check proposed solutions. $$6 x+14=2 x-2$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -4\).

Step 1: Simplify the Equation

The first thing we need to do is simplify the equation to a form where all the \(x\) terms are on one side and the constant term is on the other. We can do this as follows: \[6x - 2x = -2 - 14\] Here we subtracted \(2x\) from both sides and also subtracted 14 from both sides of the equation using the addition property of equality.

Step 2: Simplify the Coefficients

Next, simplify the coefficients on both side of the equation: \[4x = -16\] Here we simply carried out the mathematical operations on both sides.

Step 3: Solve for \(x\)

Finally, we solve for \(x\) by dividing both sides of the equation by the coefficient of \(x\) (4) using the multiplication property of equality: \[x = \frac{-16}{4}\]

Step 4: Evaluate the Solution

We then simplify the expression on the right to find the solution for \(x\): \[x = -4\]

Key Concepts

Addition Property of Equality

The Addition Property of Equality is a fundamental concept in algebra. It states that you can add or subtract the same number from both sides of an equation without changing its equality. This concept is crucial because it allows us to manipulate and simplify equations in a systematic way, making them easier to solve.

For example, in the given equation \(6x + 14 = 2x - 2\), to isolate the variables, we first need all the \(x\) terms on one side and the constants on the other. We achieve this by:

• Subtracting \(2x\) from both sides: \(6x - 2x = -2x + 2x - 2\)
• Subtracting 14 from both sides: \(6x - 2x + 14 - 14 = -2x + 14 - 2\)

This results in a simpler equation \(4x = -16\). By applying the addition property correctly, we ensure the equation remains balanced and easy to solve.

Multiplication Property of Equality

The Multiplication Property of Equality states that if you multiply (or divide) both sides of an equation by the same non-zero number, the equation remains balanced. This property is typically used to isolate the variable after rearranging terms in an equation.

In our example, after applying the Addition Property, we simplify the equation to \(4x = -16\). To solve for \(x\), we need to make \(x\) the subject of the formula by:

• Dividing both sides by \(4\), the coefficient of \(x\).

This gives us \(x = \frac{-16}{4}\).

By dividing both sides by \(4\), we ensure the equation remains balanced and we find the value of the variable. This step is a perfect example of using the multiplication property to solve for the unknown.

Equation Simplification

Equation simplification involves reducing an equation to its simplest form, where the variable is isolated on one side. This process often integrates both the addition and multiplication properties of equality.

In the original problem, simplification was achieved step-by-step:

• First, rearrange terms to move all \(x\) terms to one side (using the Addition Property).
• Then, isolate the variable \(x\) by reducing the coefficients via the Multiplication Property.

Equation simplification makes it easier to identify and solve for the variable \(x\). Each step builds logically towards a solution, ensuring that all changes preserve the original equality.

By simplifying our equation from \(6x + 14 = 2x - 2\) to \(x = -4\), we show the logical reduction process. Simplification not only allows us to find the solution but also teaches problem-solving strategies in a structured and clear manner.