Question

Verify that the given value is a solution of the given equation. $$ \frac{1}{3} x+\frac{4}{3}=2, x=2 $$

Short answer

Question: Verify if x = 2 is a solution to the equation $$\frac{1}{3}x + \frac{4}{3} = 2$$. Answer: Yes, x = 2 is a solution to the equation.

Step-by-step solution

Write down the given equation and the given x value

We have the equation $$\frac{1}{3}x + \frac{4}{3} = 2$$ and the given value of x is 2.

Substitute the given x value into the equation

Replace x with 2 in the equation: $$\frac{1}{3}(2) + \frac{4}{3}$$

Perform multiplication

Multiply the fraction by the value of x (which is 2 in this case): $$\frac{2}{3} + \frac{4}{3}$$

Add the fractions

Since the denominators are the same, we can add the numerators directly: $$\frac{2 + 4}{3}$$

Simplify the fraction

Add the numerators and simplify the fraction, getting: $$\frac{6}{3}$$

Reduce the fraction to simplest form

Reduce the fraction by dividing both numerator and denominator by their greatest common divisor, which is 3 in this case: $$\frac{6 ÷ 3}{3 ÷ 3} = 2$$

Verify that the left side of the equation equals the right side

We simplified the left side of the equation to 2, and the right side of the equation is also 2, which means the given x value (x = 2) is indeed a solution to the equation.

Key concepts

Equation Solving

Solving equations is like solving a puzzle. You find the value of a variable that makes the equation true. For example, in the equation \( \frac{1}{3} x + \frac{4}{3} = 2 \), we are looking for the value of \( x \) that makes the left side equal to the right side.

In this exercise, we're given a value for \( x \) and asked to check if it solves the equation. This involves substituting the given \( x \) value into the equation and performing arithmetic operations to see if the equation holds true. Equation solving often involves similar steps: breaking down complicated expressions, substituting values, and manipulating both sides to check for equality.

Fractions

Fractions are a way to represent parts of a whole. They consist of a numerator (top number) and a denominator (bottom number). Understanding how to work with fractions is crucial, especially in algebra.

In our problem, we deal with fractions \( \frac{1}{3} x \) and \( \frac{4}{3} \) on the left side of the equation. To process these, a crucial step is ensuring the denominators are the same, which allows us to directly add or subtract the numerators. In this case, both fractions have denominators of 3, which makes the arithmetic straightforward. Manipulating fractions properly helps to simplify the equation and find the correct value for \( x \).

Simplification

Simplification is about making an expression easier to understand or work with. In our equation, once we substitute \( x = 2 \), we end up with the expression \( \frac{2}{3} + \frac{4}{3} \).

Since both fractions have the same denominator, we can add them together: \( \frac{2 + 4}{3} = \frac{6}{3} \). This addition makes it simpler. Simplifying fractions further involves reducing them to their simplest form. Here, \( \frac{6}{3} \) can be reduced by dividing both the numerator and the denominator by their greatest common divisor, 3, which results in the integer 2.

Verification of Solutions

Verification is the process of checking if a proposed solution is indeed correct. After simplifying the equation, we ended up with \( 2 = 2 \). This verification process confirms whether the left-hand side equals the right-hand side of the equation.

In our case, since \( 2 \) matches the right side of the equation, we can conclude that \( x = 2 \) is indeed the correct solution. Verification is a vital final step in equation solving as it ensures accuracy and confirms the validity of the solution. Proper verification rules out arithmetic errors and affirms the structure of the equation.