Problem 14 - Elementary Algebra Within Reach

Question

Justify each step of the solution. $$ \begin{aligned} -2 x-8 &=0 \\ -2 x-8+8 &=0+8 \\ -2 x &=8 \\ \frac{-2 x}{-2} &=\frac{8}{-2} \\ x &=-4 \end{aligned} $$

Step-by-Step Solution

Answer

The given sequence of steps correctly follows the mathematical rules of solving a simple linear equation. Each step adheres to either the additive property (Step 2) or multiplicative property (Step 3) of equality. The solution obtained, i.e., x = -4, is accurate and justified.

Step 1: Original Equation

The initial equation presented is '-2x - 8 = 0', which is a simple linear equation.

Step 2: Applying Additive Property of Equality

The next step in the process is to get rid of the '-8' on the left side of the equation by adding '8' on both sides of the equation. So, '-2x - 8 + 8 = 0 + 8' transforms into '-2x = 8'. This is the principle of additive property, anything done on one side of equation should be done to the other side as well.

Step 3: Applying Multiplicative Property of Equality

Now, to isolate 'x', -2 (the coefficient of x) should be divided from both sides, so the equation '(-2x) / -2 = 8 / -2' simplifies to 'x = -4'. This exemplifies the multiplicative property, the multiplication or division on one side of equation should be done to the other side as well.

Key Concepts

Additive Property of Equality

In solving equations, especially linear ones, it's crucial to ensure the equations stay balanced. The additive property of equality helps us maintain this balance. When you are faced with an equation like \[ -2x - 8 = 0 \]and you need to remove the '-8' from the left side, you add '8' to both sides:\[ -2x - 8 + 8 = 0 + 8 \]This step uses the additive property of equality. Essentially, it means whatever you add to one side, you must add to the other side to keep your equation balanced. Always remember these points:

• This property keeps the expressions on both sides equal.
• It's your go-to technique for removing constants from one side to isolate the variable.

This idea of keeping things balanced is like making sure both ends of a seesaw are level, ensuring your solution process isn't skewed.

Multiplicative Property of Equality

Once you've applied the additive property and simplified your equation, you may need to divide or multiply to isolate the variable further. This is where the multiplicative property of equality comes in. For instance, after simplifying to:\[ -2x = 8 \]You divide both sides by '-2' to solve for 'x':\[ \frac{-2x}{-2} = \frac{8}{-2} \]This property allows you to multiply or divide both sides of an equation by the same number (except zero), ensuring the balance of the equation. Here are key points:

• This is crucial when the variable is multiplied or divided by a number.
• It efficiently isolates the variable, making it possible to find its value.

This property is like using the same weight on both sides of a balance scale to maintain equilibrium during manipulation.

Linear Equation Solutions

Solving linear equations is like unlocking a puzzle where the objective is to find the value of the variable that makes the equation true. The equation we solved, \[ -2x - 8 = 0 \]is a linear equation. Here's the process to reach the solution of such equations:

• Identify the variable you need to isolate (in this case, 'x').
• Use the additive property to eliminate constants from the side of the variable.
• Apply the multiplicative property to undo coefficients attached to the variable.
• Simplify the equation to find the solution.

Finally, the solution for 'x' in this problem is:\[ x = -4 \]This systematic approach ensures accuracy every time you solve a linear equation. Remember, each step builds upon the last, guiding you to the correct value for the variable.