Solving Systems of Equations in Word Problems: A Step-by-Step Guide

Systems of equations can be encountered in various real-life scenarios, from simple sums of numerical values to more complex problems involving relationships between different quantities. This article provides a detailed step-by-step approach to solving such problems, illustrated with examples and a focus on understanding the underlying principles.

Introduction to Systems of Equations

A system of equations is a set of algebraic equations involving the same set of variables. These equations are often derived from word problems, where certain relationships between variables need to be analyzed and solved. The method discussed here will walk you through the process of translating a word problem into mathematical equations and then solving those equations to find the values of the unknowns.

Example 1: Two Numbers with Given Sum and Difference

Let's consider a classic example: 'The sum of two numbers is 10. If five times the smaller number is subtracted from the larger number the result is 20. How do you find the two numbers?' To solve this, we can set up a system of equations based on the information given.

Setting Up the Equations

We will use the variables x and y, where x represents the larger number and y represents the smaller number. From the problem, we have the following equations:

The sum of the two numbers is 10: x y 10 If five times the smaller number is subtracted from the larger number the result is 20: x - 5y 20

Step-by-Step Solution

Solve the first equation for x: x 10 - y Substitute x in the second equation: 10 - y - 5y 20 Simplify and solve for y: -6y 20 - 10 which simplifies to -6y 10. Solving for y, we get y -frac{10}{6} -frac{5}{3}. Find x using the value of y: x 10 - frac{5}{3} frac{30}{3} - frac{5}{3} frac{25}{3} or approximately 8.33.

Therefore, the two numbers are approximately 8.33 (larger number) and -1.67 (smaller number).

Word Problems with Systems of Equations

Let's try another example to further illustrate the concept. If we have two numbers X and Y such that (XY 80) and (X - 3Y 16), we can solve the system step-by-step as well:

Example 2: Two Numbers with Given Product and Difference

Translate the problem into equations: From the first sentence, we can write X middot; Y 80. From the second sentence, we can write X - 3Y 16. Solve one of the equations for X: From the second equation, we get X 3Y 16. Substitute the expression for X in the first equation: (3Y 16)Y 80. Expand and simplify: 3Y2 16Y - 80 0. Solve the quadratic equation: By factoring or using the quadratic formula, we get Y 4 and X 32.

Therefore, the numbers are 32 and 4.

Conclusion

Solving systems of equations from word problems involves careful translation into mathematical equations and strategic problem-solving. By practicing with various examples, students can develop the skills necessary to tackle complex real-life problems confidently.