Problem 52 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ -\frac{3}{4} x+5=\frac{1}{4} x-3 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -16\).

Step 1: Get Rid of the Fractions

First, multiply all terms by 4 to eliminate the fractions: \[ -4*-\frac{3}{4}*x+4*5=4*\frac{1}{4}*x-4*3 \] This gives the new equation: \[ 3x+20=x-12 \].

Step 2: Group x terms

Second step is to move all terms involving x to one side and the constants to the other side. You can do this by subtracting \(x\) from both sides and subtracting 20 from both sides: \[ 3x - x = -12 - 20 \]. This can be simplified to: \[ 2x = -32 \].

Step 3: Find x

To isolate x, divide both sides by 2: \[ x = \frac{-32}{2} \]. This yields the solution: \[ x = -16 \].

Key Concepts

Eliminating Fractions in Equations

Understanding how to efficiently manage fractions within algebraic equations is fundamental in simplifying and solving these mathematical problems. To eliminate fractions, a common technique involves multiplying every term in the equation by the least common denominator (LCD) - which is simply the smallest number that each of the denominators can divide into evenly.

Take, for example, the equation \( -\frac{3}{4}x + 5 = \frac{1}{4}x - 3 \). The denominators here are 4. By multiplying every term by 4, the least common denominator, these fractions are eliminated, leaving a simpler equation with integers only \( 3x + 20 = x - 12 \). It's like clearing the clutter before organizing your room — eliminate fractions first, thus setting the stage for an easier problem-solving process.

Here's why this trick is so effective:

• It transforms the equation into one that is often easier to work with.
• By removing the fractions, you decrease the chance of making mistakes in calculation.
• It allows you to focus more on algebraic manipulations without worrying about the added complexity of fractions.

Isolating Variables

Algebra often involves finding the value of unknown variables, such as in the equation \( 3x + 20 = x - 12 \). Isolating the variable means manipulating the equation so that the unknown variable is by itself on one side of the equal sign, making its value 'visible.'

In the case of our equation, subtracting \(x\) from each side allows us to group the variable terms together \( 2x = -32 \), and subtracting 20 from each side shifts the constant terms together. This process is essential because it paves the way to get a clearer view of what the variable represents, much like finding a key piece of a puzzle that helps you see the whole picture. Here's why isolating the variable is key:

• It gives a direct answer to what the variable equals.
• Understanding the process aids in solving more complex equations in the future.
• It practices the fundamental algebraic skill of maintaining balance in an equation, a principle that whatever you do to one side, you must do to the other.

Algebraic Expressions

In algebra, an expression is a sentence containing numbers, variables, and operation signs but without an equal sign. In contrast, an equation like \( 2x = -32 \) has an equal sign and shows that two expressions are equal. Simplifying algebraic expressions is a pivotal skill, laying the groundwork for more advanced topics in mathematics.

Understanding algebraic expressions allows us to comprehend relationships between quantities and to describe these relationships succinctly. Consider the original expression before solving \( -\frac{3}{4}x + 5 \), and after eliminating fractions \( 3x + 20 \). Both represent the same quantity but in different forms. Grasping expressions and their simplifications helps with:

• Identifying patterns and making predictions based on those patterns.
• Building a strong foundation for calculus and other higher-level math courses.
• Developing critical thinking skills that are highly applicable outside mathematics, in fields such as engineering, physics, and economics.