Problem 34 - Introductory Algebra for College Students

Question

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} 2 x+4 y=5 \\ 3 x+6 y=6 \end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations has no solution.

Step 1: Rewrite the system

Let's rewrite the system of equations: \(2x+4y=5\) and \(3x+6y=6\) in standard form Ax + By = C

Step 2: Multiplying the equations

Multiply the first equation by \(3\) and the second equation by \(2\), to make the coefficients of \(x\) the same in both equations. This gives us: \(6x+12y=15\) and \(6x+12y=12\)

Step 3: Subtract the equations

We subtract the second equation from the first, giving: \(0=3\). This is a false statement, which means the original system of equations has no solution.

Key Concepts

System of Equations

A system of equations is a set of two or more equations with a same set of unknowns, where the goal is to find the values of those unknowns. In other words, we want to find the intersection of these equations represented on a graph.

When using the addition method, which is also known as the elimination method, the key is to eliminate one variable by adding or subtracting the equations from each other. This method involves three main steps:

• Writing each equation in standard form, with all variables on one side and constants on the other
• Multiplying the equations, if necessary, so the coefficients of one of the variables in both equations match
• Adding or subtracting the equations to eliminate one variable and solve for the other

This can lead to three possible outcomes:

• One solution: There's a single point where both equations intersect
• No solution: The equations never intersect, indicating parallel lines
• Infinite solutions: Equations are the same line, thus intersecting at every point on that line

No Solution

When we say a system of equations has no solution, it signifies that the lines representing these equations on a graph never cross. They are parallel to each other and have the same slope but different y-intercepts. Thus, they will never meet at a common point.

In the context of the addition method, we can identify no solution by manipulating the equations to eliminate variables. If this process results in a false statement, such as a non-zero constant equaling zero (like '0=3'), then the system of equations does not share a point of intersection and therefore has no solution. It is crucial to identify false statements to avoid wasting time attempting to solve unsolvable systems.

Set Notation

In algebra, set notation is a way of expressing a collection of objects, numbers, solutions. It's an efficient language for depicting groups of solutions and their relationships. Set notation is enclosed in curly braces \( \{ \} \).

When dealing with systems of equations, set notation is used to present the solution set succinctly. For instance, the solution to a system of equations with one solution would be given as a pair of numbers representing the point of intersection, like \( \{ (x, y) \} \). If the system has no solution, as in the given problem, we use set notation to reflect this by writing \( \{ \} \) or \( \emptyset \) to signify the empty set. If there are infinitely many solutions, the notation would include a description or a series of points that satisfy the equations.