Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{r} 2 x-3 y=1 \\ -6 x+9 y=4 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has no solution because the lines are parallel.

Step 1: Solve one equation for one variable

To use substitution, we first solve one of the equations for one of the variables. Let's start with the first equation: \\( 2x - 3y = 1 \).\Solve for \( x \): \Add \( 3y \) to both sides: \\( 2x = 3y + 1 \).\Divide both sides by 2 to solve for \( x \): \\( x = \frac{3y + 1}{2} \).

Step 2: Substitute into the other equation

Now that we have \( x \) in terms of \( y \), substitute \( x = \frac{3y + 1}{2} \) into the second equation: \\( -6x + 9y = 4 \) becomes \\( -6\left(\frac{3y + 1}{2}\right) + 9y = 4 \).

Step 3: Simplify and solve for y

Simplify the equation: \Multiply \(-6\) times \(\frac{3y + 1}{2}\): \\(-6 \frac{3y + 1}{2} = -3(3y + 1) = -9y - 3\).\Substitute back into the equation: \\(-9y - 3 + 9y = 4\).\This simplifies to \(-3 = 4\), which is not possible.\This step reveals the equations are inconsistent.

Step 4: Identify parallel lines

Since substitution leads to a contradiction, the system of equations does not have a solution. Both lines have the same slope because if you simplify, they both correspond to \( y = \frac{2}{3}x + \text{(some constant)} \), making them parallel and ensuring no intersection points.

Key Concepts

Using the Method of Substitution

The method of substitution is a handy technique to solve systems of linear equations. It involves solving one of the equations for one variable and then substituting this expression into the other equation. By doing so, you reduce a system of equations into a single equation with just one variable. This makes it easier to solve.
Think of it as a way to eliminate one variable, making the problem simpler.

• First, pick one of the equations and solve it for one of its variables.
• Next, replace or "substitute" this expression in place of the variable in the other equation.
• Finally, solve the resulting equation, and once the value of one variable is found, use it to find the value of the other variable.

In the provided exercise, we solved for \( x \) in the first equation and substituted this into the second equation to attempt to solve for \( y \). However, instead of finding a solution, we encountered a contradiction, indicating something interesting about the lines represented by these equations.

Understanding Parallel Lines in Context

Parallel lines are an essential concept in geometry and algebra. When two lines are parallel, they run in the same direction and never meet, no matter how far they are extended in either direction. This is closely linked to the idea of slope.
In terms of equations, if two lines are parallel, they will have the same slope. This means the coefficients of \( x \) and \( y \) in the linear equations will be proportional. For the system in the exercise, both lines have a slope of \( \frac{2}{3} \), confirming they are parallel.

• Parallel lines have equal slopes.
• They have different y-intercepts, which means they will never cross each other.
• An essential implication in solving systems of equations is that parallel lines indicate there is no solution, as there are no points of intersection.

Understanding this helps in identifying situations where the equations lead to no solutions, as was the case with the given exercise.

Recognizing an Inconsistent System

An inconsistent system of equations is one where no set of values for the variables will satisfy all the equations simultaneously. It means there is no common solution.
When solving such systems algebraically, especially through substitution or elimination, you might end up with a statement like \( -3 = 4 \), which is obviously false. This contradiction is a clear sign of an inconsistent system.

• Inconsistent systems are characterized by parallel lines in graphical representation.
• The equations have the same slope but different y-intercepts, thus never intersecting.
• The lack of a solution reflects the lines' inability to meet at any point on a graph.

In our step-by-step solution, the inability to solve for \( x \) or \( y \) authentically resulted from the equations representing parallel lines, making it an inconsistent system with no solution.