Problem 15 - 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. $$\left\\{\begin{array}{l} 3 x-14 y=6 \\ 5 x+7 y=10 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(\{(2, 0)\}\).

Step 1: Arrange the Equations

Initially, the given system of equations is as follows: \(3x - 14y = 6\) and \(5x + 7y = 10\).

Step 2: Multiply to Make Coefficients Equal

Before applying the addition method, it is necessary to make the coefficients of \(y\) equal in both equations. This can be achieved by multiplying the second equation by 2. This results in the following system of equations: \(3x - 14y = 6\) and \(10x + 14y = 20\).

Step 3: Apply Addition Method

Add the two equations together. This results in: \(13x = 26\).

Step 4: Solve for x

Solving for \(x\), you find that \(x = 2\).

Step 5: Substitute x into First Equation

Substitute \(x = 2\) into the first equation: \(3 * 2 - 14y = 6\), it simplifies to \(6 - 14y = 6\).

Step 6: Solve for y

Solving for \(y\), you find that \(y = 0\).

Step 7: Express Solution in Set Notation

The solution to the system of equations in set notation is \(\{(2, 0)\}\). This indicates that there is a single, unique solution to the system of equations.

Key Concepts

Addition Method in Algebra

The addition method, also known as the elimination method, is an effective way to solve a system of linear equations. It involves adding or subtracting the equations with an aim to eliminate one of the variables, making it simpler to solve for the other.

The process begins by ensuring that one pair of corresponding variables in the system of equations has the same or opposite coefficients. This can be achieved by multiplying one or both of the equations by appropriate numbers. Once the coefficients are matched, adding the equations will eliminate one of the variables. Following this, you can solve the remaining single-variable equation for its value.

To demonstrate the addition method, consider an example where the two equations from the system are:

• \(3x - 14y = 6\)
• \(5x + 7y = 10\)

By adjusting the second equation through multiplication, you get identical or opposite coefficients for the variable you intend to eliminate. Subsequently, adding the modified equations together allows you to solve for one variable and then the other, leading to the final solution.

System of Linear Equations

A system of linear equations consists of two or more linear equations that are solved simultaneously. Each linear equation represents a line in the coordinate plane, and the solution to the system corresponds to the point(s) where the lines intersect.

Solutions can vary:

• No solution occurs when the lines are parallel, indicating the equations are inconsistent.
• A unique solution exists when the lines intersect at exactly one point.
• An infinite number of solutions occur when the lines coincide, meaning one equation is a multiple of the other.

In the context of the provided exercise, we dealt with a system of two linear equations with a unique intersection point. Therefore, after applying the addition method correctly, we were able to find one specific solution to the system.

Set Notation for Solutions

Set notation provides a standard way to present solutions, especially when working with system of equations. In set notation, solutions are written as ordered pairs \(x, y\) enclosed within curly braces \( \{ \} \).

If there is a unique solution, it will be represented as a single ordered pair within the set. An example based on our system is \( \{ (2, 0) \} \), showing the point where the two lines cross. In situations where there are no solutions or infinitely many solutions, set notation can still be used:

• No solution is often denoted by \( \emptyset \), the empty set symbol.
• Infinite solutions may be represented by describing all the points lying along a line, or by representing the solution as a set containing an equation or parameter that describes the solutions.

By using set notation, we can succinctly communicate the outcome of solving a system of equations, making it clear and precise for anyone analyzing the solution.