Problem 40 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$-3 x-6=-36$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 10\).

Step 1: Isolate the Variable Term

To solve the equation \(-3x - 6 = -36\), first, we need to isolate the term containing the variable. Start by adding 6 to both sides of the equation to remove the constant term from the left side. This gives us: \[-3x - 6 + 6 = -36 + 6\]. Simplifying this results in: \[-3x = -30\].

Step 2: Solve for the Variable

With the equation now \(-3x = -30\), divide both sides by \(-3\) to solve for \(x\). This results in: \[x = \frac{-30}{-3}\]. Simplifying the fraction gives \(x = 10\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding the value of the unknown variable that makes the equation true. The ultimate goal is to get the variable on one side and constants on the other. This often requires performing operations on both sides of the equation. A balanced approach keeps the equation true. In our exercise, we have an initial equation of \(-3x - 6 = -36\). We need to apply basic algebraic principles, like the addition property of equality, to unravel the equation step by step. This ensures we do not disrupt the balance of the equation.Remember, every change we make must keep both sides equal. Performing the same mathematical operation on both sides is crucial to maintain this balance.

Isolate the Variable

To solve an equation, isolating the variable is an essential step. This means manipulating the equation so that the variable is by itself on one side of the equals sign.

Addition Property of Equality

In the example \(-3x - 6 = -36\), our first move is to eliminate the \(-6\) from the left side. We do so by using the addition property of equality. Adding \(6\) to both sides of the equation gives:
\(-3x - 6 + 6 = -36 + 6\)
Simplifying, we achieve:
\(-3x = -30\).By removing the constant, we are one step closer to isolating the variable \(x\). This simplification is key to finding the solution in the next steps.

Algebraic Steps

Algebraic steps are logical moves performed to solve equations, simplifying expressions andworking towards finding the variable's value. For every equation, these steps must be carefully chosen and executed to reach the correct solution.

Dividing to Solve

Once we have the variable term isolated as in \(-3x = -30\), the next algebraic step is to solve for \(x\). Here, this requires dividing both sides by \(-3\):
\(x = \frac{-30}{-3}\)
Upon simplifying the fraction, we find:
\(x = 10\).These steps, often repetitive across different equations, reinforce the practice of balancing both sides until the variable is unveiled. Always check your final answer to ensure it's correct by plugging it back into the original equation.