Problem 61 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. In a film, the actor Charles Coburn played an elderly "uncle" character criticized for marrying a woman when he is 3 times her age. He wittily replies, "Ah, but in 20 years time I shall only be twice her age." How old is the "uncle" and the woman?

Step-by-Step Solution

Answer

The woman is currently 20 years old, and the uncle is 60 years old.

Step 1: Set up the equations

Let the woman’s current age be \(x\), and the uncle's current age be \(y\). From the given information, we can make the following two equations: \(y = 3x\) (since the uncle is three times the woman's age) and \(y + 20 = 2(x + 20)\) (since in 20 years, the uncle will be twice the woman's age).

Step 2: Solve the equations

Solve the equations for \(x\) and \(y\). Since \(y = 3x\), substitute \(y\) in the second equation with \(3x\): \(3x + 20 = 2(x + 20)\). Simplify this equation to get \(x = 20\). So, the woman is 20 years old at present. Substitute \(x\) into the first equation to solve for \(y\): \(y = 3 * 20 = 60\). Hence, the uncle is 60 years old at present.

Step 3: Confirm the solution

Confirm that the solution fits the given problem. The uncle is indeed three times the age of the woman. And in twenty years, the uncle will be 80 and the woman will be 40, at which point he will be twice her age. So the solution fits the problem.

Key Concepts

Linear Equations

Understanding linear equations is essential when solving algebra word problems. A linear equation is an equation that can be written in the form \( ax + b = 0 \), where \( a \) and \( b \) are constants and \( x \) is a variable. The graph of a linear equation in two variables is a straight line.

When tackling word problems, the key is to translate the given narratives or statements into linear equations. This involves identifying numerical information and the relationships between unknown quantities. For instance, if one quantity is three times another, as in the provided exercise, we can capture this relationship with the equation \( y = 3x \).

Thereafter, the equations need to be manipulated algebraically to find the values of the unknowns. This might include simplifying expressions, combining like terms, and using substitution or elimination methods to solve for the variables—skills that are foundational to mastering algebra.

Age-Related Algebra Problems

Age-related algebra problems often involve setting up relationships between the ages of two or more individuals at different points in time. In the example regarding 'uncle' and the woman, we are dealing with a scenario that requires projecting ages into the future, which is a common aspect of these types of problems.

To navigate age-related algebra problems, it is crucial to clearly define the variables representing the current ages of the individuals involved. Use logical statements to express the given age relationships. In our exercise, the fact that the uncle will be 'only twice' the woman's age in 20 years gave us a second linear equation, \( y + 20 = 2(x + 20) \).

An understanding of how to set up these equations and solve them is valuable not only in academic contexts but also in real-world scenarios where predicting ages may be relevant, such as planning for retirement or understanding demographic changes.

System of Equations

When faced with multiple linear equations involving the same variables, as often is the case in age-related problems, we're dealing with a system of equations. A system of equations requires us to find a common solution—that is, a set of values for the variables that satisfy all the equations simultaneously.

In solving a system of equations, there are several methods at our disposal, including graphing, substitution, elimination, and matrix operations. The substitution method, used in our exercise example, involves solving one of the equations for one variable in terms of the other and then substituting that expression into the other equation.

For the 'uncle' scenario, after expressing the second equation as \( 3x + 20 = 2(x + 20) \) and solving for \( x \), we substituted back to find the value of \( y \). This use of the substitution method aligns the abstract process of solving equations with a tangible and identifiable outcome, reinforcing the real-world application of these mathematical concepts.