Problem 43 - Algebra 1: Concepts and Skills

Question

Copy the solution steps shown. Then write an explanation for each step in the right-hand column. Solution Step \(\frac{5 x}{2}+3=6\) \(\frac{5 x}{2}=3\) \(5 x=6\) \(x=\frac{6}{5}\)

Step-by-Step Solution

Answer

The solution to the equation \(\frac{5 x}{2}+3=6\) is \(x=\frac{6}{5}\).

Step 1: Identify and Work on the Equation

Start with the original equation \(\frac{5 x}{2}+3=6\). The goal is to isolate x by moving all other terms to the other side of the equation.

Step 2: Subtract 3 from both sides of the equation

Simplify the equation by subtracting 3 from both sides of the equation to isolate \(\frac{5x}{2}\) on one side. This results in \(\frac{5 x}{2}=3\).

Step 3: Multiply both sides of the equation by 2

Multiply both sides of the equation by 2 to eliminate the fraction. This gives \(5x=6\).

Step 4: Divide both sides of the equation by 5

Divide both sides of the equation by 5 to finally isolate x. After division, we get \(x=\frac{6}{5}\).

Key Concepts

Algebraic Equations

An algebraic equation is a mathematical statement that expresses the equality of two algebraic expressions. It consists of variables, coefficients, and constants. The variable in an equation represents an unknown value that we are trying to find. In our exercise, the equation we are dealing with is \( \frac{5 x}{2}+3=6 \). The first step in solving this algebraic equation is to understand the relationship between the terms. Each term plays a role and understanding how to manipulate these terms to isolate the variable is the key to finding the solution.

Algebraic equations are fundamental in various fields of science and engineering, serving as a tool to model real-world scenarios. By learning how to solve these equations, students can apply their knowledge to complex problems and develop analytical skills that are essential in higher education and professional careers.

Isolation of Variables

The process of isolating a variable involves manipulating an equation to place the variable we're interested in by itself on one side of the equation, making it the 'subject' of the formula. This step is crucial because it turns the statement into a form where the value of the variable can be easily determined. In our example, isolating the variable x requires us to perform operations that will eliminate other terms while keeping the equation balanced.

To achieve this, we use inverse operations: subtraction is used to cancel addition, and division is used to nullify multiplication. This is demonstrated in the exercise where we first subtract 3 from both sides to remove the constant term, and then multiply by 2 to remove the fraction. Finally, dividing by 5 isolates x, giving us \(x=\frac{6}{5}\). Understanding and mastering the isolation of variables is crucial for students to solve equations efficiently.

Fraction Operations

Fraction operations are another fundamental concept within algebra that allows us to manipulate equations that involve fractions. The goal is to simplify the equation or isolate the variable. Our exercise involves the steps for ridding the equation of the fraction. Initially, by multiplying both sides of the equation by 2, we effectively clear the denominator of the fraction \(\frac{5x}{2}=3\), as multiplying by 2 is the inverse operation of dividing by 2.

This manipulation gives us \(5x=6\), a simpler equation without fractions. The next important operation with fractions is division. When we divide both sides by 5, we are actually multiplying by the reciprocal, which leaves us with the solution \(x=\frac{6}{5}\). Understanding how to work with fractions in equations is vital, as it simplifies complex problems and allows for quicker and easier solutions.