Problem 46 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$-7 x=-3 x-8$$

Step-by-Step Solution

Answer

Our proposed solution, \(x = -2\), does not satisfy the equation \(-7x = -3x - 8\). The actual solution seems to be a mistake.

Step 1: Eliminate the variable on one side

To begin with, add 7x to both sides of the equation which will help eliminate the variable on one side. Then the equation becomes \(4x = -8\).

Step 2: Solve for x

To solve for \(x\), divide both sides of the equation by 4. That gives \(x = -2\).

Step 3: Check the solution

Now, substitute \(x = -2\) into the original equation. This helps to verify if the solution is valid. A correct solution should balance the equation, making the left side equal to the right side. The equation thus becomes \(-7*(-2) = -3*(-2) - 8\), which simplifies to \(14 = 6 - 8\), which is not true. Therefore, the proposed solution \(x = -2\) is not a valid solution to the initial equation.

Key Concepts

Addition Property of Equality

The Addition Property of Equality is a fundamental concept in solving equations, which states that you can add the same number to both sides of an equation without affecting its balance. It maintains the equation's equality while making it easier to manipulate and simplify. For example, in the equation \(-7x = -3x - 8\), we can add \(7x\) to both sides to eliminate the variable from one side and simplify the equation:

• Subtract \(-3x\) as \(+7x\) on both sides.
• In this case, the equation changes to \(4x = -8\).

This step clears away variables from one side, which simplifies solving the equation. Always remember, whatever operation you perform on one side, you must do to the other. This property is invaluable for setting up the equation into a form that's easier to solve.

Multiplication Property of Equality

The Multiplication Property of Equality lets you multiply or divide both sides of an equation by the same non-zero number to keep the equation balanced. It is particularly useful for isolating the variable to find its value.
In our simplified equation, \(4x = -8\), we divide both sides by 4 to solve for \(x\):

• Divide both sides by 4.
• The solution becomes \(x = -2\).

By executing the same operation on both sides, the equality of the equation is preserved, and you can then clearly identify the value of the unknown variable.
This step is essential because it neatly isolates the variable and provides a direct solution. Use this property whenever you need to "undo" multiplication or division affecting the variable.

Checking Solutions

Checking your solution is a critical step in solving equations and ensures the found solution actually satisfies the original problem. After calculating, it's important to verify by substituting the solution back into the original equation.
Consider our proposed solution, \(x = -2\). Plug this value back into the original equation \(-7x = -3x - 8\):

• Substitute \(-2\) for \(x\): \(-7(-2) = -3(-2) - 8\).
• Simplify both sides: \(14 = 6 - 8\).
• However, \(14\) does not equal \(-2\).

Since both sides of the equation do not equate, \(x = -2\) is not a correct solution for the equation. This step is mandatory as it not only confirms the accuracy of your solution but also reinforces your understanding of the process. It safeguards against any errors you might have made during the calculations.