Problem 28 - College Algebra

Question

In \(2001,\) the most populous countrics in the world were China and India. In that year, China's population exceeded India's by 260 million. Combined, the two countries had a population of 2310 million. Determine the 2001 population for China and India.

Step-by-Step Solution

Answer

In 2001, China's population was 1285 million and India's population was 1025 million.

Step 1: Write out the equations

First, let's define C as China's population and I as India's population (both in millions). From the problem, we obtained two equations: \(C = I + 260\) and \(C + I = 2310\)

Step 2: Substitution

We can use the substitution method to solve this system of equations. We take the equation \(C = I + 260\) and substitute \(C\) into the second equation: \((I + 260) + I = 2310\).

Step 3: Solve for I (India's population)

Combine like terms to get \(2I + 260 = 2310\). Then subtract 260 from both sides to get \(2I = 2050\). Finally, divide both sides by 2 to solve for I: \(I = 1025\). This means India's population was 1025 million in 2001.

Step 4: Solve for C (China's population)

Substitute \(I = 1025\) into the first equation \(C = I + 260\), we get \(C = 1025 + 260 = 1285\). So, China's population was 1285 million in 2001.

Key Concepts

Substitution Method

The substitution method is an efficient technique for solving a system of equations. It's particularly useful when one equation *can easily be solved for one variable in terms of the others*. In our exercise, we were given the system:
\[C = I + 260\]
\[C + I = 2310\]
To apply the substitution method, we start by isolating one variable. In this case, the variable 'C' was already expressed in terms of 'I'. This isolated equation was then substituted into the other equation, which reduced the system to a single variable equation.
When you're substituting, be mindful to

• Substitute accurately, to not introduce errors into the equation.
• Ensure every instance of the isolated variable is replaced in the second equation.
• Simplify the resulting equation to solve for the remaining variable.

By substituting and eliminating one variable, you simplify the problem and pave the way to finding the solution for both variables, leading to the population numbers for China and India in the exercise.

Algebraic Equations

Algebraic equations are the foundation of algebra and are used to represent real-world problems, like the population comparison problem we have here. An *algebraic equation is a statement of equality involving variables and constants*. It's made up of terms, which can consist of variables (like 'C' and 'I' for the populations of China and India), constants (like the '260 million' difference in population), and coefficients (in our case, the implied '1' before 'C' and 'I').
Key aspects of algebraic equations to remember include:

• The need to perform the same operation on both sides to maintain equality.
• The goal of isolating the variable to find its value.
• The importance of checking your solution by substituting it back into the original equation to ensure it holds true.

In solving real-world problems, crafting an algebraic equation correctly is crucial as it represents the problem's conditions and determines the approach for finding a solution.

Solving Linear Equations

Solving linear equations is a process we use to find the value of variables that make the equation true. *A linear equation is an equation where each term is either a constant or the product of a constant and a single variable.* The distinction 'linear' means that the variable is not raised to any power other than one.
The procedure for solving linear equations involves several steps:

• Simplifying each side of the equation, such as removing parentheses and combining like terms.
• Moving all terms containing variables to one side and all constants to the other, aiming to isolate the variable.
• Dividing or multiplying to get the variable alone and solve the equation.

In our population problem, after substituting, we got a linear equation \(2I + 260 = 2310\), which we simplified by subtracting 260 and dividing by 2, to find India's population. This process is fundamental in algebra and forms the basis for solving more complex mathematical problems.