Problem 12 - Intermediate Algebra

Question

For Problems \(1-18\), use the elimination-by-addition method to solve each system. (Objective 1 ) $$ \left(\begin{array}{l} 2 x+3 y=-5 \\ 4 x+6 y=10 \end{array}\right) $$

Step-by-Step Solution

Answer

The system has no solution due to inconsistency.

Step 1: Review the System

We are given a system of linear equations: \(2x + 3y = -5\) and \(4x + 6y = 10\). Notice that the coefficients of the second equation are exactly twice the coefficients of the first equation.

Step 2: Simplify the System

Divide the entire second equation by 2 to simplify: \(\frac{4x}{2} + \frac{6y}{2} = \frac{10}{2}\) which results in \(2x + 3y = 5\). Now our system is \(2x + 3y = -5\) and \(2x + 3y = 5\).

Step 3: Compare the Equations

Notice that both equations now have the same left side but different right sides. This indicates an inconsistency in the system.

Step 4: Identify the Inconsistency

Subtract one equation from the other: \[(2x + 3y) - (2x + 3y) = (-5) - (5)\]. This results in \(0 = -10\), which is a contradiction.

Step 5: Conclude with No Solution

Since \(0 = -10\) is false, the system is inconsistent, meaning there are no values for \(x\) and \(y\) that satisfy both equations simultaneously.

Key Concepts

Understanding a System of Equations

A system of equations is a set of two or more equations with the same variables. These equations are often used to determine the values of unknown variables that satisfy all equations in the system simultaneously. In simpler terms, it means finding a common solution for two or more mathematical statements. These systems can be expressed in various ways, such as through matrices or algebraic expressions.

• Each equation in the system represents a line when graphed on a coordinate plane.
• The solution to the system is the point or points where these lines intersect.
• Systems can have one solution, infinite solutions, or no solution at all.

Understanding systems of equations is fundamental in algebra as it applies to a wide range of real-world problems, from economics to engineering.

Identifying an Inconsistent System

When working with a system of equations, an important classification to understand is whether the system is consistent or inconsistent. An inconsistent system is one that has no solution. This occurs when the lines represented by the equations are parallel and never intersect.

• Parallel lines in 2D space never meet. Hence, the system they represent has no common solution.
• In the given example, after simplifying both equations, it becomes obvious that they are different yet have the same left-hand side, indicating they are two parallel lines.
• Checking for consistency involves simplifying the equations, as shown in the problem, to see if any contradictions arise, such as "0 = -10" which is clearly false.

If a system is inconsistent, it means that there is no set of values that will satisfy all equations simultaneously, as shown in our example.

Recognizing Linear Equations

Linear equations are the simplest form of equations and can be easily recognized. They have variables raised to the power of one (no squares, cubes, etc.), take the form of a straight line when graphed, and look like either of the following:

• Standard form: Ax + By = C
• Slope-intercept form: y = mx + b
• Point-slope form: y - y₁ = m(x - x₁)

The key aspect of linear equations is their predictability and simplicity.

• They represent straight lines in a two-dimensional space.
• The coefficients represent slopes and intercepts, which determine the position and angle of the line.

These equations form the backbone of solving many algebraic problems, including systems of equations.

Steps in Solving Systems of Equations

Solving systems of equations can be achieved through various methods, with the elimination method being one of the most efficient. Here's a streamlined process for using elimination to solve systems:

• Step 1: Arrange the equations so that corresponding sides are aligned (e.g., Ax + By = C).
• Step 2: Manipulate the equations so that one of the variables has the same coefficient. This often involves multiplying one or both equations by certain constants.
• Step 3: Add or subtract the equations from one another to eliminate one variable. Simplify the remaining equation to find the value of one variable.
• Step 4: Substitute back to solve for the second variable.
• Step 5: Check the solution by substituting both variables back into the original equations to verify they hold true.

In the provided example, the system was simplified to reveal it was inconsistent. This highlighted an essential aspect of solving systems: not all sets of equations have a solution, particularly when they represent parallel lines.