Problem 89 - College Algebra with Modeling and Visualization

Question

Solve the equation for the specified variable. $$ y=3(x-2)+x \text { for } x $$

Step-by-Step Solution

Answer

The solution for \(x\) is \(x = \frac{y + 6}{4}\).

Step 1: Distribute in the Equation

Begin by applying the distributive property to the expression within the equation. The expression given is \(3(x - 2)\). Distribute the \(3\) to both terms inside the parenthesis: \(3(x - 2) = 3x - 6\).

Step 2: Simplify the Equation

Substitute the distributed expression back into the equation: \(y = 3x - 6 + x\). Combine like terms on the right-hand side: \(3x + x = 4x\), so the equation simplifies to \(y = 4x - 6\).

Step 3: Isolate the x-term

To isolate the \(x\)-term, first move the constant to the other side of the equation by adding \(6\) to both sides: \(y + 6 = 4x\).

Step 4: Solve for x

Divide both sides of the equation by \(4\) to solve for \(x\): \(x = \frac{y + 6}{4}\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps you break down more complex expressions. It essentially states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately and then adding the results. For the expression \(3(x - 2)\), the distributive property allows us to rewrite it by distributing or spreading the \(3\) across both terms inside the parentheses.

• Multiply \(3\) by \(x\), which gives \(3x\).
• Multiply \(3\) by \(-2\), resulting in \(-6\).

Now, putting it back together, the expression \(3(x - 2)\) simplifies to \(3x - 6\). This simplification is crucial because it removes parentheses, making the equation easier to handle in subsequent steps.

Isolating Variables

Isolating variables is a key step in solving equations, especially when you want to solve for a particular variable like in this exercise for \(x\). It involves manipulating the equation in such a way that the variable of interest stands alone on one side of the equation.

In our example, after using the distributive property, we obtain the equation \(y = 4x - 6\). To isolate the \(x\)-term, we aim to have it by itself on the right side. We start by getting rid of any constants present along with the variable:

• Add \(6\) to both sides to offset the \(-6\), which gives us \(y + 6 = 4x\).

These systematic steps help in clearing the path to solve for \(x\) easily and ensures we maintain the equality of the original equation.

Solving Equations

Solving equations is the process of finding the value of the variable(s) that make the equation true. After isolating \(x\), the equation becomes \(y + 6 = 4x\). This is where we solve for the variable by performing operations that maintain the balance of the equation.

To finally find \(x\):

• Divide each term by \(4\). This step results in \(x = \frac{y + 6}{4}\), successfully solving for \(x\).

Remember, when solving equations, our goal is to simplify and isolate the variable of interest step by step, using operations like addition, subtraction, multiplication, and division. Each operation must be applied to both sides of the equation to keep it balanced. This systematic approach ensures the accuracy and consistency needed in algebra.