Problem 12 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ 3+\frac{1}{3} x=5 $$

Step-by-Step Solution

Answer

The solution is \(x = 6\).

Step 1: Remove the Fraction

Multiply every term of the equation by 3 in order to eliminate the fraction: \[3 \times 3 + 3 \times \frac{1}{3}x = 3 \times 5\] This results in: \[9 + x = 15\]

Step 2: Isolate the Variable

Subtract 9 from both sides of the equation to solve for \(x\):\[9 + x - 9 = 15 - 9\]This simplifies to:\[x = 6\]

Key Concepts

Solving Equations

In mathematics, solving equations is all about finding the value of the unknown variable that makes the equation true. This process can sometimes seem tricky, but with a systematic approach, it can actually be quite straightforward.

To solve an equation, you have to perform operations that simplify the expression, such as adding, subtracting, multiplying, or dividing both sides of the equation. It is important that whatever you do to one side, you must also do to the other side to maintain balance, just like a scale.

In our exercise, the given equation is linear, meaning its graph forms a straight line, and it doesn't involve any squared variables or higher powers. The solution to a linear equation typically involves basic arithmetic or algebraic manipulations that gradually simplify the equation until you can clearly see the solution.

Isolating Variables

Isolating variables is a crucial step in solving equations. It involves getting the variable by itself on one side of the equation to determine its value.

In the exercise, after eliminating the fraction by multiplying everything by 3, we had the equation:

• \(9 + x = 15\)

Here, isolating the variable \(x\) required subtracting 9 from both sides:

• \(9 + x - 9 = 15 - 9\)

This subtraction isolates \(x\) because we subtract the number that is on the same side as \(x\) which results in:

• \(x = 6\)

The idea here is to "zero out" what is on the same side as the variable so that the variable stands alone.

Algebraic Manipulation

Algebraic manipulation involves the rearrangement or simplification of algebraic expressions using a consistent set of operations and techniques. These techniques are designed to make finding solutions easier.

In our given example, the major manipulation was eliminating the fraction by multiplying every term by 3. This step simplifies the equation by removing complex fractions, which are often a bumbling block.

Structured manipulation included moving terms from one side of the equation to another, such as subtracting 9 from \(9 + x = 15\). This follows the principle of balancing, ensuring every operation has an equal and opposite reaction on the equation.

Effective algebraic manipulation is a toolkit for successfully solving equations. It helps in transforming complex expressions into simpler forms without changing the actual solutions. This adaptability is key to working through different types of equations, not just linear ones.