Problem 65 - Algebra A Combined Function

Question

\(-2 x-\frac{1}{2}=\frac{7}{2}\)

Step-by-Step Solution

Answer

The solution is \(x = -2\).

Step 1: Eliminate the Fraction

To simplify the equation, eliminate the fraction. Multiply every term in the equation by 2, which is the denominator of the fractions.The original equation is:\[-2x - \frac{1}{2} = \frac{7}{2}\]Multiply each term by 2:\[2(-2x) - 2\left(\frac{1}{2}\right) = 2\left(\frac{7}{2}\right)\]This gives:\[-4x - 1 = 7\]

Step 2: Isolate the Variable Term

Add 1 to both sides of the equation to move the constant to the right side:\[-4x - 1 + 1 = 7 + 1\]This simplifies to:\[-4x = 8\]

Step 3: Solve for x

Divide both sides by -4 to solve for x:\[x = \frac{8}{-4}\]So, we find:\[x = -2\]

Key Concepts

Solving Equations

Solving equations is a foundational skill in algebra. Suppose you have an equation, a mathematical statement indicating that two expressions are equal. Your goal is to find the values of the variables that make the equation true.
In simpler terms, solving an equation is like finding the secret number that makes both sides of the equation balance perfectly. You can think of an equation as a scale where both sides must weigh the same. If you change something on one side, you must do the same to the other to keep it balanced.
When solving equations:

• Always perform the same operation on both sides.
• Check your solution by substituting the variable back into the original equation.
• Be patient and careful with each step to avoid mistakes.

Fractions in Equations

Handling fractions in equations might seem tricky at first, but with practice, it becomes easy. Fractions indicate division, and dealing with them usually means finding a way to make the equation less complex. In the given exercise, the presence of fractions makes the equation appear more complicated.
To eliminate fractions:

• Identify the denominators in the fractions.
• Multiply every term in the equation by the least common denominator (LCD) to clear fractions.

This step transforms the equation into a simpler form, making it easier to solve. For instance, the exercise required multiplying every term by 2 to clear a fraction of 1/2, leading us to a simpler equation without any fractions.

Isolation of Variables

Isolating variables is crucial for solving equations. It involves rearranging the equation to have the variable you are solving for on one side by itself. This process resembles peeling away layers to uncover what's inside.

After removing fractions, focus on the terms involving the variable. Use basic operations, like addition or subtraction, to move terms around. For example:

• Subtract constant terms from both sides to move them away from the variable.
• Add terms to both sides as necessary to keep the equation balanced.

In the exercise, adding 1 to both sides helped isolate the term with the variable, creating a straightforward path to solving the equation.

Algebraic Manipulation

Algebraic manipulation is the art of rearranging and simplifying equations to make them solvable. It uses various mathematical operations, adhering to algebra rules, to find the unknown variable.

When dealing with algebraic manipulation, keep these tips in mind:

• Follow the order of operations to avoid mistakes: PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).
• To solve for a variable, perform inverse operations to both sides to counteract an operation affecting the variable.

In the last step of the exercise, dividing both sides by -4 ensures that the variable 'x' stands alone. Mastering these techniques makes algebra much easier and helps you tackle more complex problems in the future.