Problem 13 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using the multiplication property of equality. Be sure to check your proposed solutions. $$-8 x=6$$

Step-by-Step Solution

Answer

So, the solution to the equation -8x = 6 is \(x = -0.75\). Check your solution by plugging the value of x back into the original equation to verify that it holds true.

Step 1: Validate the original equation

First, make sure the equation is valid. The original equation:-8x = 6 is well defined.

Step 2: Apply the multiplication property of equality

To find the value of x, adhere to the multiplication property of equality. This requires division (or multiplication with the reciprocal) of both sides by the same number to maintain the equation's balance. Here, we divide both sides by -8.

Step 3: Perform the operation

Divide both sides of the equation by -8, \(-8x ÷ -8 = 6 ÷ -8\). As a result, the equation simplifies to \(x = -0.75\)

Step 4: Check the solution

Now, check whether the solution is correct or not. Substitute the value of x in the original equation. If both sides of the equation are equal, then the solution is correct. So, \(-8*(-0.75) = 6\) which proves that the solution is correct.

Key Concepts

Solving Linear Equations

Understanding how to solve linear equations is key to success in algebra. A linear equation is one that forms a straight line when graphed, and it typically looks something like this: \( ax + b = c \), where \( a \) and \( b \) are coefficients and \( c \) is a constant. The goal is to isolate the variable, \( x \), on one side of the equation.

To tackle an equation like \( -8x = 6 \) we aim to get \( x \) by itself. Since \( x \) is multiplied by \( -8 \) in this instance, we can reverse this operation by dividing both sides by \( -8 \) (as seen in our Step 3). This action utilizes the multiplication property of equality, which suggests that if we do the same thing to both sides of the equation, it remains balanced and true.

The result is \( x = -0.75 \), which simplifies the equation. Always remember to verify your solution by plugging it back into the original equation, ensuring that both sides equal each other. For our exercise, once \( x \) is confirmed, we've solved the linear equation correctly.

Equation Balancing

Equation balancing is like a seesaw; both sides must remain equal for it to stay balanced. Algebra is built on this crucial principle. No matter how complex an equation, you must perform equivalent operations on both sides to maintain its balance. This is where our multiplication property of equality comes into play.

With our example \( -8x = 6 \), dividing by \( -8 \) on both sides doesn't change the equality, just the form of the expression, ultimately helping us identify that \( x = -0.75 \). It's a systematic approach to changing the equation's appearance without altering its intrinsic value. Checking your solution by back-substituting into the original equation confirms the balance has been maintained I just like you would test a balanced seesaw by ensuring both sides are level.

Algebraic Properties

Algebraic properties are the rules that govern the manipulation of equations. These properties ensure that our operations are valid and that the equation's integrity is kept intact. One such property is the multiplication property of equality, vital when we face equations like \( -8x = 6 \).

This property allows us to divide each side by \( -8 \), because if two quantities are equal, then their ratios to the same non-zero quantity are also equal. This concept is deeply rooted in the principle of equivalent operations that are pivotal for the manipulation of equations. Other algebraic properties include the distributive property, the commutative property, and the associative property, each ensuring that different algebraic steps produce valid results and follow a logical, systematic process.

Adhering to these properties allows us to arrive at a valid solution confidently and ensures our algebraic manipulations lead to a correct solution.