Problem 33 - 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. $$\left\\{\begin{aligned} 7 x-3 y &=4 \\ -14 x+6 y &=-7 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system of equations has no solution.

Step 1: Multiply the equations

The aim is to eliminate one variable. Looking at the coefficients of \(x\) and \(y\) in both equations, they can easily be eliminated by multiplication. So, multiply the first equation by 2 and second equation by 1.

Step 2: Add the multiplied equations

The result of the multiplication is:\[14x - 6y = 8\] \[-14x + 6y = -7\] These are then added together to get:\[14x - 14x - 6y + 6y = 8 - 7 \Rightarrow 0x = 1\]

Step 3: Interpret the result

Since \(0x = 1\) is impossible (0 multiplied with any real number cannot equal 1), it implies that there are no solutions for this system of equations. This typically means that the two lines represented by the equations are parallel and do not intersect.

Key Concepts

Addition Method

The addition method is a systematic way to solve a system of linear equations. It involves modifying the equations so that adding them cancels out one of the variables, making it easier to solve the system.

• Step 1: Adjust the coefficients of one variable in both equations by multiplying each equation by suitable numbers. This aligns the coefficients, enabling you to eliminate a variable.
• Step 2: Add the resulting equations. One variable should entirely cancel out, allowing you to solve for the remaining variable.
• Step 3: If one variable is eliminated, solve for the other variable. If no variable remains, interpret the resulting mathematical expression.

In the provided exercise, we eliminated variable "y" by multiplying each equation to align their coefficients. Addition canceled them, simplifying the problem significantly.

Set Notation

Set notation is a versatile way to express the solution of systems of equations. It is often used to present solutions succinctly and precisely.

• Typically, solutions are written in curly braces, like \({x,y}\).
• The solution set may contain pairs of values that satisfy all equations simultaneously.
• If there is no solution, the set can be denoted as an empty set, \(\emptyset\).

In this context, since our system has "no solution," we use the empty set to communicate that no pair of values can satisfy both equations at the same time. This denotes that the system is inconsistent with no intersection point.

Linear Equations

Linear equations are equations of the first order, represented graphically by straight lines. They have no exponents higher than one.

• The general form is \(ax + by = c\), where "a," "b," and "c" are constants.
• Solutions are found where these lines intersect, which represents the point where both equations are satisfied simultaneously.

In the exercise, we had two linear equations. Their coefficients suggested a possibility of being parallel with no intersection point, leading to the conclusion that they have no common solution.

No Solution

A system of equations may have "no solution" if the lines are parallel. This means the two lines never intersect on a graph.

• If you simplify a system of equations to a false statement such as \(0x = 1\), it indicates no solution.
• This situation is called an 'inconsistent system', meaning the equations contradict each other.

In the given problem, after applying the addition method, the resulting equation implies inconsistency. Thus, the lines do not meet at any point, verifying no solution exists for this system.