Question

If \(5+n=9-\frac{1}{3} n\), what is the value of \(n\) ? A) 3 B) 4 C) 6 D) \(\frac{21}{2}\)

Short answer

A) 3

Step-by-step solution

Add \(\frac{1}{3}n\) to both sides of the equation

To isolate the variable \(n\), we need to add \(\frac{1}{3}n\) to both sides of the equation. This will eliminate the term \(-\frac{1}{3}n\) from the right-hand side of the equation. \[5 + n + \frac{1}{3}n = 9 - \frac{1}{3}n + \frac{1}{3}n\]

Simplify the equation

Now we need to simplify the equation by combining the terms with the variable \(n\). \[\frac{1}{3}n + n = \frac{1}{3}n + \frac{3}{3}n = \frac{4}{3}n\] So, the simplified equation becomes: \[5 + \frac{4}{3}n = 9\]

Subtract 5 from both sides of the equation

Next, we need to subtract 5 from both sides of the equation to further isolate the variable \(n\). \[\frac{4}{3}n = 9 - 5\]

Simplify the equation again

Now we can simplify the right side of the equation: \[\frac{4}{3}n = 4\]

Divide by \(\frac{4}{3}\) to solve for \(n\)

Finally, we need to divide both sides of the equation by \(\frac{4}{3}\) to solve for the value of \(n\). \[n = \frac{4}{\frac{4}{3}}\]

Simplify the fraction to find the value of \(n\)

To simplify the fraction, multiply the numerator and the denominator by the reciprocal of the denominator: \[n = \frac{4}{1} \times \frac{3}{4}\] This simplifies to: \[n = 3\] So, the value of \(n\) is 3, which corresponds to choice A.

Key concepts

Solving Equations

Equations are mathematical statements that assert the equality of two expressions. Solving equations involves finding the value of the variable that makes the equation true. In the exercise given, we are tasked with determining the value of \( n \) in the equation \( 5 + n = 9 - \frac{1}{3}n \). The process involves manipulating the equation to isolate the variable \( n \).

One of the fundamental techniques in solving equations is performing the same operation on both sides of the equation. This might include adding, subtracting, multiplying, or dividing both sides by the same number or expression. This approach ensures that the equality of the equation is maintained while progressively isolating the variable. For example, in step 1 of our solution, we add \( \frac{1}{3}n \) to both sides of the equation to eliminate the term \(-\frac{1}{3}n\) from one side.

Once the equation is simplified to a form that clearly screams one solution, you will be able to solve for \( n \) when \( \frac{4}{3}n = 4\), by performing division. This final step gives us the value of \( n \) as 3. Understanding these steps and the logic behind them is key to mastering equation-solving in algebra.

Linear Equations

Linear equations are equations that graph as straight lines when plotted on a coordinate plane. These equations are typically in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. They are called 'linear' because their solutions don't involve any exponents other than 1, making them straight lines in graphical representation.

The exercise example, \( 5 + n = 9 - \frac{1}{3}n \), is a classic example of a linear equation. Even though it appears slightly more complex due to the fraction \(-\frac{1}{3}n\), the solution still involves basic operations to bring the terms involving \( n \) together.

• Firstly, rearrange the terms to group \( n \) terms on one side. This offers a clearer picture of the linear relationship.
• After simplification, which involves the consolidation of terms, the equation \( \frac{4}{3}n = 4 \) is obtained.

The simplicity of a linear equation is what makes it one of the most fundamental mathematical concepts, serving as a building block for more complex algebraic equations.

SAT Math Preparation

The SAT Math section is designed to gauge a student's ability to solve problems using mathematical concepts. Mastery of linear equations is often critical for scoring well in this segment, as problems typically revolve around fundamental algebraic operations and equation-solving methods.

Here’s how understanding and mastering the solution to equations like \( 5 + n = 9 - \frac{1}{3}n \) prepares you for the SAT Math section:

• Test Familiarity: You become familiar with the typical types of problems featured, which often include linear equations.
• Time Management: Knowing how to effectively isolate and solve for variables helps in managing time efficiently during the test.
• Confidence Boost: Practicing different types of algebra problems boosts your confidence, ensuring a calm and composed test-taking experience.

By applying logical mathematical techniques and understanding why each step is taken, you develop critical thinking skills essential for the SAT and beyond. It's not just about getting the right answer but understanding the process that leads to it.