Problem 33 - College Algebra

Question

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(y=3 x-5\) \(21 x-35=7 y\)

Step-by-Step Solution

Answer

The system of equations has infinitely many solutions represented by the set: \{(x,y) | y = 3x - 5, x is a real number\}

Step 1: Solve the Second Equation for y

Isolate y in the second equation to get it in the form y = mx + b. This can be done by dividing every term by 7: \(y = 3x - 5\)

Step 2: Compare the Equations

Now that both equations are in the same form, compare them. It can be noticed that both equations are identical. So, the lines coincide and the system has infinitely many solutions.

Step 3: Express the Solution Set in Set Notation

The solution to the system is all the points (x,y) that satisfy both equations. Since the lines coincide, the solution set includes all points on either line. In set notation, it can be written as \{(x,y) | y = 3x - 5, x is a real number\}

Key Concepts

Infinite Solutions

When solving systems of equations, we may encounter a peculiar situation where the system doesn't just have one solution, but rather an infinite number of solutions. This occurs when the equations represent the same line; that is, they are identical when graphed. In the given exercise, after isolating 'y' from the second equation and comparing it to the first, we discover that both equations are, in fact, the same: \( y = 3x - 5 \).

This tells us that every point on the line \( y = 3x - 5 \) is a solution to the system. Thus, there isn't a single intersection point, but the entire line represents the set of solutions. Understanding this concept is crucial for students to grasp why sometimes a system doesn't have a limited number of solutions but an endless array.

Set Notation

Set notation is a systematic way of describing a set of numbers or points that comprise the solutions to an equation or system. In our solved exercise, the set of solutions is expressed as \( \{(x,y) | y = 3x - 5, x \text{ is a real number}\} \). This notation is read as 'the set of all points (x,y) such that y equals 3x minus 5, and x is a real number.'

Utilizing set notation not only makes the description of the solution set more precise, but it also helps in understanding the nature of the solutions. For instance, in this case, it emphasizes that the solution is not just a single pair of values but every possible pair that satisfies the equation, with 'x' being any real number.

Method of Substitution

The method of substitution is a powerful technique for solving systems of equations. It involves rearranging one of the equations to express one variable in terms of the other—like isolating 'y' in the example—as we have done in Step 1 of the solution. Once a variable has been isolated, it can be substituted into the other equation, effectively reducing the system to a single equation with one variable.

In the case of finding infinite solutions, applying the method of substitution would lead us to an identity – an equation that is true for all values of the variable – confirming that the system has an infinite number of solutions. Thus, mastering the method of substitution is integral for students, as it not only helps in finding specific solutions but also in identifying systems with infinite solutions.

Comparing Equations

Sometimes the quickest way to understand the relationship between two linear equations is to directly compare them. This can be done by rewriting the equations in the same form, such as the slope-intercept form \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept. If, upon comparing, the equations have the same slope and y-intercept, as in the given exercise, it is evident that they represent the same line.

Comparing equations is a vital skill in algebra, as it allows quick identification of complex relationships, like finding out if two lines are parallel, coincident, or intersect at a point. In conclusion, mastering the art of comparing equations is another fundamental tool in a student's mathematical toolkit and serves as a gateway to understanding different outcomes in a system of equations.