Problem 39 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The solution to the equation -x - 3 = 3 is x = -6.

Step 1: Simplify By Adding 3 To Both Sides

To get rid of the negative sign attached to x, and to start isolating the term x, add 3 on both sides of the equation. This is achieved by using the addition property of equality, which states that if you add the same number to both sides of the equation, the equality is still maintained. Therefore, when you add 3 to both sides, the equation transforms from -x - 3 = 3 to -x = 6.

Step 2: Multiply by -1 To Solve for x

Results from step 1 indicates '-x = 6'. Now to solve for x, multiply both sides of the equation by -1 (multiplication property of equality). Doing this cancels out the negative sign in front of the x on the left side of the equation, leading to the solution x = -6.

Step 3: Check the Proposed Solution

Substitute the solution obtained (x = -6) into the original equation (-x - 3 = 3) to verify. If the left-hand side equals the right-hand side after substitution, then the solution is correct. Substitute x in '-x -3' with -6, the result is 3, which equals to the right-hand side of the equation. Hence, the solution obtained (x = -6) is correct.

Key Concepts

Addition Property of Equality

This property is essential for maintaining equality while solving equations. It states that you can add the same number to both sides of an equation without changing its balance. In our original exercise \(-x - 3 = 3\), the goal is to isolate the variable \(x\). To begin with, we add \(3\) to both sides. This is because subtracting \(3\) from each side would make one side negative. By adding \(3\), the equation becomes \(-x = 6\). The addition property simplifies the equation, making it easier to focus solely on the variable part.

Multiplication Property of Equality

Once the addition property has been applied, the multiplication property of equality can be used to deal with coefficients or negative signs. This property allows you to multiply both sides of an equation by the same non-zero number without affecting the equality. In the equation \(-x = 6\), the goal is to make \(x\) positive. Therefore, multiply each side by \(-1\), which changes the equation to \(x = -6\). The multiplication property of equality is crucial in flipping signs or scaling equations to simplify them, and it helped us to find the solution for \(x\) in this problem.

Checking Solutions

After finding a proposed solution in algebra, it's always good practice to verify it by substituting it back into the original equation. This ensures that the solution satisfies the equation, confirming its correctness. For our problem, the solution proposed is \(x = -6\). By substituting \(x = -6\) back into the original equation \(-x - 3 = 3\), we replace \(x\) with \(-6\) and simplify it: \(-(-6) - 3\) becomes \(6 - 3\), which equals \(3\). Since both sides of the equation are equal, the solution is verified as correct. Checking solutions acts as a safety net ensuring that no mistakes were made in earlier steps.