Problem 35 - Algebra and Trigonometry

Question

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. \(\left\\{\begin{array}{l}{3 x-2 y=-5} \\ {4 x+y=8}\end{array}\right.\)

Step-by-Step Solution

Answer

The system of linear equations has exactly one solution, which is \(x = 1, y = 4\).

Step 1: Preparation

First, simplify both equations, if needed. In this case, both equations are already in standard form (Ax + By = C), so there's no need for any simplification.

Step 2: Begin the elimination

Modify equation 1 to eliminate one of the variables when added to equation 2. Here, multiply equation 2 by 2 to match with the same coefficient of y in equation 1. The new system of equations is as follows: \[ \left\{\begin{array}{l}{3 x-2 y=-5} \ {8 x+2 y=16}\end{array}\right. \]

Step 3: Eliminate y

Add equation 1 and the modified equation 2. This will eliminate y: (3x - 2y) + (8x + 2y) = -5 + 16. Simplifying this gives \(11x = 11\).

Step 4: Solve for x

Solve the resulting equation for x by dividing both sides by 11: \(x = 1\).

Step 5: Substitute x into the second equation

Substitute \(x = 1\) into the second equation (\(4x + y = 8\)) to solve for y: \(4*1 + y = 8 \Rightarrow y = 8 - 4 = 4\).

Step 6: Verify the solution

The last step is to check the values of x and y in both equations. If both are true, then \(x = 1, y = 4\) is the solution for this system of equations. So, check: for equation 1 (\(3*1 - 2*4 = -5\)) and for equation 2 (\(4*1 + 4 = 8\)). As you can see, both equations are correct, which confirms the solution.

Key Concepts

Elimination Method

The elimination method is a way to solve systems of equations, where the goal is to eliminate one of the variables by adding or subtracting the equations. When one variable is eliminated, it becomes easier to solve for the remaining variable. Then, back-substitute the found value into one of the original equations to find the other variable.

Here's how it works:

• First, align the equations in a way that allows one variable to cancel out when the equations are added or subtracted. This may require multiplying one or both equations by a suitable number to match coefficients.
• Add or subtract the equations to eliminate one variable. Only one should remain, making the system easier to solve.
• Solve the resulting single equation for the remaining variable.
• Use the found value in one of the original equations to solve for the other variable.

By changing the coefficients of the variables, the elimination method simplifies the system down to a single-variable equation, which is typically straightforward to solve.

Standard Form of Equation

In algebra, the standard form of a linear equation is given by the expression \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. This form is incredibly useful when applying methods like elimination or substitution to solve systems of equations.

Why is the standard form preferred? Here are a few reasons:

• It is a consistent format that makes equations easier to compare and manipulate.
• It provides a clear layout for rearranging or combining equations, which is essential in the elimination method.
• The coefficients \(A\) and \(B\) directly reveal critical information about the line, such as its slope and y-intercept.

In the given exercise, both equations were already in standard form, which streamlined the process of applying the elimination method.

No Solution Systems

A system of equations has no solution when the lines represented by the equations are parallel. This means the lines never meet, and thus there is no set of coordinates that satisfies both equations.

Identifying no solution systems involves:

• Checking that the ratio of the coefficients of \(x\) and \(y\) are equivalent, but the constants on the other side of the equations don't match.
• Parallel lines are characterized by having the same slope, yet different y-intercepts, confirming they will not intersect.

For example, consider the equations \(2x - 3y = 6\) and \(4x - 6y = 12\). They simplify to the same line, leading to no intersection, and therefore, no solutions.

Infinitely Many Solutions Systems

A system has infinitely many solutions when the equations represent the same line. This occurs because every point on the line satisfies both equations, leading to countless solutions.

Key characteristics of systems with infinitely many solutions are:

• Equations that simplify to be identical, meaning they have proportional coefficients and the same constant term.
• The lines overlap completely, indicating every point on one line is also on the other.

Consider the equations \(x + y = 3\) and \(2x + 2y = 6\). By simplifying the second equation, it becomes clear both equations describe the identical line. Thus, they have infinitely many solutions.