Problem 34 - Introductory Algebra for College Students

Question

Solve each equation and check your proposed solution in Exercises. Begin your work by rewriting each equation without fractions. $$\frac{3 x}{4}-9=-6$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\).

Step 1: Remove the Fraction

In order to remove the fraction, multiply every term on both sides by 4. It gives \(3x-36=-24\).

Step 2: Isolate the Variable

The next step is to get the variable on one side by adding 36 to both sides. This leads to the equation: \(3x=12\). Divide every term by 3 to solve for \(x\). This gives \(x=4\).

Step 3: Check the Solution

In order to check if the solution is correct, substitute \(x=4\) into the original equation. The equation becomes: \(\frac{3*4}{4} - 9 = -6\). Simplifying the expression gives \(3 - 9 = -6\), which is a true statement. Therefore, the solution is valid.

Key Concepts

Algebra

Algebra is an essential branch of mathematics that deals with symbols and the rules for manipulating these symbols. In algebra, letters and symbols, like \( x \) or \( y \), are used to represent numbers in equations and formulas. This allows us to formulate and solve real-world problems where certain quantities are unknown. When solving linear equations in algebra, the primary goal is to find the value of the variable that makes the equation true.

In the provided exercise, we encountered a linear equation that involves one variable, \( x \). The process of solving it involved rearranging and simplifying the equation to find \( x \). A strong foundation in algebra helps in simplifying these equations efficiently. Mastering algebra is crucial as it forms the basis for advanced mathematical concepts and applications.

Fractions

Fractions represent a part of a whole and are an integral aspect of mathematics. They consist of a numerator and a denominator. The numerator indicates how many parts you have, while the denominator indicates how many parts make up a whole.

In solving equations that involve fractions, like the one in our exercise, a common approach is to eliminate the fractions to simplify the calculation. This is typically done by multiplying every term by the least common multiple of the denominators involved.

• In the exercise, we multiplied every term by 4 to get rid of the fraction \( \frac{3x}{4} \).
• After removing the fraction, the equation became easier to handle and solve for \( x \).

Learning how to manipulate fractions, especially in algebraic equations, can greatly simplify the solving process and lead to quicker solutions.

Equations

An equation is a mathematical statement that asserts the equality of two expressions. Solving an equation involves finding the value(s) of the variable(s) that make the statement true.

In linear equations, each term is either a constant or the product of a constant and a single variable. Consider the equation from the exercise: \( \frac{3x}{4} - 9 = -6 \). We solved this equation step by step to isolate \( x \):

• First, we eliminated the fraction by multiplying each term by 4.
• Next, we rearranged terms to bring all variable terms to one side and constants to the other.
• Finally, we divided by the coefficient of the variable to solve for \( x \).

Understanding how to manipulate and solve equations is a fundamental skill in algebra, enabling students to solve various types of real-world problems.

Checking Solutions

Verifying that your solution to an equation is correct is a crucial and final step in solving any equation.

After finding a solution, substitute it back into the original equation to ensure it satisfies the equation.

• This involves replacing the variable with the solution found; in our case, substituting \( x = 4 \) back into \( \frac{3x}{4} - 9 = -6 \).
• Check that both sides of the equation are equal; solving confirmed the left side simplified to \(3 - 9\), which indeed equaled \(-6\).

By checking solutions, we confirm the accuracy of our answer and ensure no steps or miscalculations were overlooked. This habit enhances problem-solving skills and confidence in handling mathematical challenges.