Problem 64 - Understanding Elementary Algebra with Geometry

Question

A seamstress can hem a skirt in 10 minutes and make the cuffs on a pair of pants in 15 minutes. On a certain day she has allotted 300 minutes to doing these tasks. (a) Let \(s\) represent the number of skirts she hems and \(p\) represent the number of pairs of pants she cuffs. Write an equation that reflects the given situation. (b) Sketch the graph of this relationship. Be sure to label the coordinate axes clearly. (c) If she hems 18 skirts, use the equation you obtained in part (a) to find the number of pairs of pants she cuffs.

Step-by-Step Solution

Answer

Equation: \( 10s + 15p = 300 \), If she hems 18 skirts, she can cuff 8 pairs of pants.

Step 1: Define Variables

Let s represent the number of skirts hemmed and p represent the number of pairs of pants cuffed. The time taken to hem a skirt is 10 minutes, and the time taken to cuff a pair of pants is 15 minutes.

Step 2: Formulate the Equation

Since the total time allotted is 300 minutes, write an equation that sums up the time taken for both tasks. The equation is: \[ 10s + 15p = 300 \]

Step 3: Sketch the Graph

To sketch the graph, first express p in terms of s from the equation obtained. \[ 15p = 300 - 10s \]\[ p = \frac{300 - 10s}{15} \]Plot the line on a coordinate plane with s on the x-axis and p on the y-axis. Identify the intercepts by setting s and p to zero respectively..If s=0, then \[ p = 20 \]If p=0, then \[ s = 30 \] So, plot the points (0, 20) and (30, 0) and draw the line that connects them.

Step 4: Calculate Pants Cuffed for 18 Hemmed Skirts

Substitute s = 18 into the equation from part (a) to find p .\[ 10(18) + 15p = 300 \]\[ 180 + 15p = 300 \]\[ 15p = 120 \]\[ p = 8 \]So, if she hems 18 skirts, she can cuff 8 pairs of pants.

Key Concepts

Linear Equations

In algebra, a linear equation is an equation that can be graphically represented as a straight line.
The general form of a linear equation is given by: \(ax + by = c\) where \(a, b,\) and \(c\) are constants.
In the exercise provided, we considered two variables, \(s\) for skirts hemmed and \(p\) for pairs of pants cuffed. The time taken for each task is added in proportions to provide a total time constraint, hence the formation of a linear equation: \( 10s + 15p = 300\) This equation states that the total time spent hemming skirts and cuffing pants cannot exceed 300 minutes. Understanding how to form these equations is essential in solving algebraic word problems efficiently.
Longer equations can be simplified into linear equations, making them easier to manage and solve.

Graphing Linear Equations

Graphing linear equations simplifies the visualization of relationships between different variables.
To graph the equation from our exercise, we first need to rearrange it to solve for one variable in terms of the other. This helps in identifying the intercepts: \(p = \frac{300 - 10s}{15}\) Here's how we proceed:
1. Set \(s = 0\) to find the \(p\)-intercept:
\(p = \frac{300 - 10(0)}{15} = 20\) 2. Set \(p = 0\) to find the \(s\)-intercept:
\(s = \frac{300 - 15(0)}{10} = 30\) 3. Plot these intercepts \((0, 20)\) and \((30, 0)\) on the graph.
Once these intercepts are plotted, they can be connected with a straight line representing the linear relationship.\( \text {For example, you will get a straight line if you plot the above points.}\)Graphing these equations is a visual aid which clarifies the maximum number of skirts and pairs of pants that she can complete within the given time.
This graph is not just a visual representation, but also an insightful tool for future planning and resource allocation.

Solving Systems of Equations

A system of equations involves solving two or more linear equations simultaneously to find a common solution.
In our exercise, although there is only one equation, the process of solving it when given a certain condition could be considered similar.
To find the number of pairs of pants cuffed when hemming 18 skirts, we solve the equation for \(p\):For instance:1. Substitute \(s = 18\) into the equation: \(10(18) + 15p = 300\)2. Simplify and solve for \(p\): \(180 + 15p = 300 , 15p = 120 , p = 8\)Systems of equations are particularly useful when dealing with multiple conflicting constraints. Solving these equations helps find the optimal solution that satisfies all constraints, ensuring effective decision-making and time management.
So, if she hems 18 skirts, she can cuff 8 pairs of pants. This balancing act between two tasks exemplifies how algebra can be wielded to tackle real-life problems with precision.