Problem 33 - Elementary Algebra

Question

Solve. $$ 13 x=-12 $$

Step-by-Step Solution

Answer

\( x = \frac{-12}{13} \)

Step 1: Identify the equation type

The equation given is a simple linear equation: \( 13x = -12 \). Our task is to find the value of \( x \).

Step 2: Isolate the variable

To solve for \( x \), we need to isolate it. We do this by dividing both sides of the equation by 13.

Step 3: Execute the division

Divide both sides of the equation by 13:\[x = \frac{-12}{13}\]

Step 4: Simplify (if possible)

In this case, \(-12/13\) cannot be simplified further, so our answer remains as \( x = \frac{-12}{13} \).

Key Concepts

Solving Equations

When tackling linear equations, such as \( 13x = -12 \), the primary goal is determining the value of the unknown variable, \( x \). Linear equations typically include one or more terms with variable coefficients. They are called "linear" because they form a straight line when graphed. To find the solution, follow these steps:

• Identify the type: Make sure it's a linear equation, which means it has the standard form \( ax = b \) or can be rearranged into it.
• Focus on isolating the variable. This often involves performing operations like addition, subtraction, multiplication, or division on both sides of the equation.
• Simplify where necessary. This helps in obtaining the most straightforward expression for the unknown variable.

Recognize that solving equations involves moving algebraic terms around while maintaining equality. The solution you find will ensure that both sides of the equation give the same value when evaluated.

Isolation of Variables

Isolating the variable in an equation is a crucial step in solving it. The aim here is to have the unknown variable on one side of the equation and everything else on the opposite side. In our example, \( 13x = -12 \), we need to solve for \( x \). This is achieved through dividing both sides by the coefficient of \( x \), which is 13.

• Begin by identifying the coefficient of the variable you want to isolate. Here, it’s 13.
• Perform the inverse operation. Since the \( x \) is being multiplied by 13, divide both sides by 13 to undo this multiplication. This yields \( x = \frac{-12}{13} \).

By completing this step, you effectively "free" the variable, allowing you to understand its exact value in terms of whatever constants remain on the other side of the equation. This process could involve more steps for more complex equations, but the principle remains the same.

Simplifying Fractions

Once you have isolated the variable and expressed it as a fraction, like in the equation \( x = \frac{-12}{13} \), the next step is to simplify the fraction if possible. Simplifying fractions helps in providing a clearer and more concise answer.In our example, \( \frac{-12}{13} \) is already in its simplest form:

• The numerator and denominator share no common factors other than 1.
• Always check whether the numerator can be factored or divided by a number greater than 1, and if the denominator allows the same division.

Sometimes, fractions can be simplified further by canceling out common factors, which makes the expression easier to interpret and aids in more complex calculations later. If no simplification is possible, like in our case, then the fraction \( \frac{-12}{13} \) is the best representation of your solution.