Problem 19 - Algebra and Trigonometry

Question

\(9- 46\) 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, expand both sides of the equation to remove the parentheses. For the left side:\[ 2(1-x) = 2 \cdot 1 - 2 \cdot x = 2 - 2x \] For the right side:\[ 3(1 + 2x) + 5 = 3 \cdot 1 + 3 \cdot 2x + 5 = 3 + 6x + 5 = 8 + 6x \]

Step 2: Set Equation

Now, write the expanded left and right expressions as a single equation:\[ 2 - 2x = 8 + 6x \]

Step 3: Add 2x to Both Sides

To get all terms involving \(x\) on one side, add \(2x\) to both sides of the equation:\[ 2 - 2x + 2x = 8 + 6x + 2x \]Simplifying gives:\[ 2 = 8 + 8x \]

Step 4: Subtract 8 from Both Sides

To isolate the term with \(x\), subtract 8 from both sides:\[ 2 - 8 = 8x \]Simplifying gives:\[ -6 = 8x \]

Step 5: Solve for x

Divide both sides by 8 to solve for \(x\):\[ x = \frac{-6}{8} = \frac{-3}{4} \]

Key Concepts

Solving Linear Equations

When we talk about solving linear equations, we simply mean finding the value of the variable that makes the equation true. Linear equations are called 'linear' because they form a straight line when graphed on a coordinate plane. These equations often appear in the form of \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable.
To solve a linear equation, follow these basic steps:

• Expand expressions to eliminate parentheses.
• Get all terms with the variable on one side of the equation.
• Isolate the variable by performing arithmetic operations.
• Check your solution by substituting it back into the original equation.

Solving equations involves careful calculation and logical reasoning to ensure that every move leads us closer to the solution.

Algebraic Manipulation

Algebraic manipulation is a process that involves rearranging and simplifying equations to find the value of unknown variables. It includes operations like addition, subtraction, multiplication, and division, and allows us to transform an equation without changing its meaning.
In the process of solving linear equations, algebraic manipulation might involve:

• Combining like terms: This means adding or subtracting terms that have the same variable part.
• Moving terms from one side of the equation to the other: This is often referred to as "transposing" terms and involves changing their sign as you move them across the equal sign.
• Factoring: Though less common in simple linear equations, this can help break down more complex expressions into simpler parts.
• Simplifying complex fractions or terms before manipulating the entire equation.

Practicing algebraic manipulation skills prepares you for solving not only linear equations but also more complicated algebraic problems.

Equivalent Equations

Equivalent equations are equations that have the same solutions. When we perform algebraic operations, we often try to maintain equivalency, meaning that if two equations are equivalent, they will have the same set of solutions.
To create equivalent equations, you can:

• Add or subtract the same number from both sides of an equation.
• Multiply or divide every term of the equation by the same non-zero number.
• Use distribution to multiply out terms.
• Simplify both sides to their simplest form.

Each transformation used in reaching the solution for the original exercise maintains equivalency. For example, in solving \(2 - 2x = 8 + 6x\), adding \(2x\) to both sides results in an equivalent equation \(2 = 8 + 8x\). Each operation preserves the balance and integrity of the equation.

Expand Expressions

Expanding expressions is a key step when solving equations involving parentheses. To expand means to multiply the number outside the parentheses by each term inside. This is an essential skill in making equations easier to manage and solve.
Using the distributive property, if you have an expression such as \(2(1-x)\), you multiply 2 by each term inside the parentheses:

• 2 multiplied by 1 equals 2.
• 2 multiplied by \(-x\) equals \(-2x\).

This gives the expanded expression \(2 - 2x\). Similarly, on the right side of the equation \(3(1 + 2x) + 5\), the distributive property gives you \(3 + 6x + 5\), which simplifies to \(8 + 6x\).
This shows the power of the distributive property for expanding expressions and simplifying equations for further steps. Practicing expansion reduces errors and enhances your ability to solve a wide range of algebraic equations effectively.