Problem 19 - Precalculus Mathematics for Calculus

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. $$2(1-x)=3(1+2 x)+5$$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{3}{4}\).

Step 1: Expand both sides

First, use the distributive property to expand both sides of the equation. Expand the left side: \[2(1-x) = 2 \cdot 1 - 2 \cdot x = 2 - 2x\] Expand the right side: \[3(1+2x)+5 = 3 \cdot 1 + 3 \cdot 2x + 5 = 3 + 6x + 5\] Combine like terms on the right side to get: \[3 + 6x + 5 = 8 + 6x\]So the equation becomes: \[2 - 2x = 8 + 6x\]

Step 2: Move variable terms to one side

To solve for \(x\), move all terms involving \(x\) to one side and constant terms to the other. Add \(2x\) to both sides:\[2 - 2x + 2x = 8 + 6x + 2x\]Simplifying both sides gives:\[2 = 8 + 8x\]

Step 3: Move constant terms to other side

Now, isolate \(x\) by moving the constant terms on one side. Subtract \(8\) from both sides:\[2 - 8 = 8x\]Simplifying gives:\[-6 = 8x\]

Step 4: Solve for x

Finally, solve for \(x\) by dividing both sides of the equation by \(8\):\[-6/8 = x\]Simplify the fraction:\[x = -\frac{3}{4}\]

Key Concepts

Distributive Property

The distributive property is a crucial concept in algebra that helps break down expressions into simpler parts. It's a rule that allows you to multiply a single term by each term inside a parenthesis. This operation ensures that each term is treated equally and the expression is expanded correctly.
For example, if you have an expression like \(a(b + c)\), the distributive property lets you expand it to \(ab + ac\). This makes complex expressions easier to handle by distributing the multiplication over the terms inside the parentheses.
In the original exercise, the distributive property was used on both sides of the equation:

• Left Side: \(2(1-x)\) was expanded to \(2 \cdot 1 - 2 \cdot x = 2 - 2x\).
• Right Side: \(3(1+2x)\) was expanded to \(3 \cdot 1 + 3 \cdot 2x = 3 + 6x\).

This initial step of distributive property helps set the stage for further simplification and solving of the equation.

Combining Like Terms

Combining like terms is an essential algebraic technique used to simplify expressions and equations by merging terms that have the same variable component. It makes equations easier to manage by reducing the number of terms and providing a clearer path to the solution.
After applying the distributive property, you'll often end up with similar variable terms or constant terms. To combine them, simply add or subtract their coefficients.
For instance, in the exercise, after expanding, the right side of the equation was:

• \(3 + 6x + 5\).
• Combining the constants \(3\) and \(5\) resulted in \(8 + 6x\).

This step simplifies the equation significantly, making it easier to isolate the variables in the subsequent steps.

Solving for x

Solving for \(x\) in an equation involves a series of steps to isolate \(x\) on one side, resulting in a clear solution. It's the primary goal when working with linear equations. The steps include moving all terms involving \(x\) to one side and constants to the opposite side to form an equation where \(x\) can be explicitly calculated.
From our example, after combining like terms, the equation was \(2 - 2x = 8 + 6x\). The next step involves rearranging:

• Adding \(2x\) to both sides to isolate the term with \(x\) gives: \(2 = 8 + 8x\).

After this, you move all constants away from the variable part, leading the path to isolate \(x\). Each step simplifies the equation, bringing it closer to its solution.

Isolating Variables

Isolating variables is the process of manipulating an equation so that the variable of interest, in this case \(x\), is alone on one side of the equation. This involves using inverse operations to systematically reduce the equation to a form where you can clearly find the value of the variable.
Looking at the final steps from the original problem:

• After rearranging the terms, the equation was: \(2 = 8 + 8x\).
• Subtracting \(8\) from both sides gives: \(-6 = 8x\).
• Dividing both sides by \(8\) isolates \(x\), resulting in \(x = -\frac{3}{4}\).

This final division step is essential, and it provides the complete solution for \(x\). Understanding isolating variables helps you solve linear equations systematically, ensuring you reach the correct answer with clarity.