Problem 8 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. \(3(x-2)=-6\)

Step-by-Step Solution

Answer

After distributing, isolating the variable, and checking the solution, we find that the solution to the equation is \(x = 0\).

Step 1: Distribute

The first step is to distribute 3 into the equation \(3(x-2)=-6\). That becomes \(3x - 6= -6\).

Step 2: Isolate the variable on one side of the equation

To isolate the variable \(x\), add 6 to both sides of the equation. Which leads to \(3x = 0\).

Step 3: Solve for the variable

Now, divide both sides by 3 to completely isolate \(x\). This gives \(x = 0\).

Step 4: Check the solution

Substitute \(x = 0\) into the original equation \(3(x-2)=-6\). This gives \(-6=-6\), which is a true statement. So, \(x = 0\) is the correct solution to the equation.

Key Concepts

Distributive Property

Understanding the distributive property is crucial when dealing with algebraic expressions, especially when they involve parentheses. It allows us to multiply a single term by each term inside the parentheses. Take our operand '3' and the expression within parentheses, '(x-2)'. By applying the distributive property, we multiply '3' by both 'x' and '-2' separately, resulting in the equation 3x - 6 = -6.

Here's a simpler look at the process: 3 * (x-2) = 3*x + 3*(-2) = 3x - 6.
Think of it like distributing slices of a pie evenly among several plates; the number '3' is the slices, and '(x-2)' are the plates.

Isolating the Variable

The goal of isolating the variable is to get the variable 'x' by itself on one side of the equation to find its value. To isolate 'x' in the equation 3x - 6 = -6, we need to eliminate '-6' from the left side. We do this by performing the same operation on both sides, which maintains the balance of the equation.

By adding '6' to each side, we neutralize '-6' on the left, leaving us with 3x = 0. Now 'x' is almost isolated. This strategy simplifies complex equations into a form that is much easier to solve, analogous to removing all the obstacles in the way of a runner, allowing them to reach the finish line unhindered.

Checking Solutions in Algebra

After finding a solution, confirming its validity is an essential step. Checking the solution involves substituting the variable in the original equation with the value obtained. In our case, with the proposed solution x = 0, we substitute 'x' with '0' into the original equation 3(x-2) = -6 to get:

3(0 - 2) = -6, which simplifies to -6 = -6. This true statement verifies our solution. It's similar to making sure a key fits the lock perfectly; just as the correct key will turn the lock effortlessly, the correct solution will satisfy the equation neatly.

Step-by-Step Algebraic Solution

A step-by-step approach to solving algebra problems can vastly improve understanding and accuracy. By breaking down the process into manageable chunks, it becomes easier to follow along and spot potential errors. Let's recap our problem's step-by-step solution:

• Distribute: Apply the distributive property to 3(x - 2) to get 3x - 6.
• Isolate the Variable: Add '6' to both sides to move all terms except for the variable to one side, resulting in 3x = 0.
• Solve for Variable: Divide both sides by '3' to solve for 'x', which gives x = 0.
• Check the Solution: Substitute 'x' back into the original equation to ensure the solution is correct.

Taking these steps one by one can be likened to following a recipe. Each step builds upon the previous one, leading to a successful outcome when followed in order.