Problem 17 - Algebra 1

Question

LANGUAGE CLASSES At Barton High School, 45 students are taking Japanese. This number has been increasing at a rate of 3 students per year. The number of students taking German is 108 and has been decreasing at a rate of 4 students per year. At these rates, when will the number of students taking Japanese equal the number taking German? Write and solve an equation to answer the question. Check your answer with a table or a graph.

Step-by-Step Solution

Answer

The number of students studying Japanese will equal the number of students studying German in 9 years.

Step 1: Formulate Equations

The equation for the Japanese class can be written as: \(J(t) = 45 + 3t\) where \(J(t)\) is the number of students at time \(t\) and \(t\) is given in years. Similarly, the equation for the German class is \(G(t) = 108 - 4t\). The goal is to find the year when the number of students will be equal in both courses, i.e. when \(J(t) = G(t)\).

Step 2: Solve for Time

Setting \(J(t) = G(t)\) gives \(45 + 3t = 108 - 4t\). Solving for \(t\) will give us the year when the number of Japanese and German students will be equal.

Step 3: Execute Algebraic Computations

Solving the equation \(45 + 3t = 108 - 4t\), combine like terms to get \(7t = 63\). Then, divide both sides by 7 to solve for \(t\), resulting in \(t = 9\).

Step 4: Result Interpretation

The result \(t = 9\) years means that in 9 years the number of students studying Japanese will equal the number of students studying German.

Key Concepts

Linear Equations

Linear equations are mathematical statements that show the equality between two expressions. They consist of variables and constants and are expressed in the form of a straight line when graphed on a coordinate plane. In our context, linear equations are used to determine how the number of students in the Japanese and German classes changes over time.

Here's how we used linear equations in our problem:

• The equation for the Japanese class: \(J(t) = 45 + 3t\).
• The equation for the German class: \(G(t) = 108 - 4t\).

The variable \(t\) represents time in years. Each equation models the number of students in the class after a certain number of years. The equations are linear because they make a straight line when plotted and involve operations like addition, subtraction, and multiplication by constants.

Rate of Change

The rate of change in a linear equation shows how much a quantity increases or decreases over time. In our exercise, it's crucial for understanding how the number of students in each class changes each year.

For the Japanese class, the rate of change is 3 students per year. This means each year, 3 more students enroll in Japanese classes. Mathematically, it's the coefficient of \(t\) in the equation \(J(t) = 45 + 3t\).

Similarly, for the German class, the rate of change is -4 students per year, reflecting a decrease. Again, it is the coefficient of \(t\) but negative, in the equation \(G(t) = 108 - 4t\).

Understanding rate of change helps predict future student numbers in both classes. When the rates are plotted, the intersections indicate when the number of students will be the same.

Problem Solving

Problem solving involves using appropriate mathematical methods to find a solution to a question. Our problem was to find out when the number of Japanese and German language students would be the same.

We began by setting up equations to model the situation:

• For Japanese: \(J(t) = 45 + 3t\)
• For German: \(G(t) = 108 - 4t\)

After forming these equations, we equated them to find \(t\), representing the number of years until the students' numbers are equal.

This type of problem-solving demands logical structuring and the application of algebraic methods to give an accurate solution.

Substitution Method

The substitution method is a fundamental technique used to solve systems of equations. However, in this problem, we solved a single equation with one unknown by executing algebraic operations.

In substitution, typically one equation is solved for one variable, and that expression is substituted into the other. Yet, here, we equated the two linear equations to find the common year when both classes had the same number of students. Doing so simplified to one equation: \(45 + 3t = 108 - 4t\).

Solving for \(t\) involved isolating the variable through:

• Combining like terms to get: \(7t = 63\)
• Solving for \(t\) by dividing by 7 to get \(t = 9\)

Although we didn't use a direct substitution of variables here, understanding this method helps recognize similar strategies in different algebraic problems.