Problem 37 - A Graphical Approach to College Algebra

Question

Solve each system by elimination. $$\begin{array}{r}9 x-5 y=1 \\\\-18 x+10 y=1\end{array}$$

Step-by-Step Solution

Answer

The system has no solution, as the lines are parallel and do not intersect.

Step 1: Set Up the Equations

We are given the following system of equations to solve by elimination:\[ 9x - 5y = 1 \]\[ -18x + 10y = 1 \].

Step 2: Simplify the Equations

Observe that the second equation can be simplified. Divide the entire second equation by -2 to simplify:\[ -18x + 10y = 1 \Rightarrow 9x - 5y = -\frac{1}{2} \].

Step 3: Compare the Equations

Now, compare the simplified equations:- First equation: \( 9x - 5y = 1 \)- Simplified second equation: \( 9x - 5y = -\frac{1}{2} \).The equations are parallel (same coefficients for x and y) but have different constant terms, making them inconsistent.

Step 4: Analyze the Solution

Since the two lines represented by these equations are parallel (have the same slope) but different y-intercepts, they do not intersect. This implies there is no solution to this system.

Key Concepts

Elimination Method

The elimination method is a useful tool for solving systems of linear equations. It involves adding or subtracting the equations in order to eliminate one of the variables.
This allows you to solve for the other variable easily.

• The first step in using the elimination method is to align your equations, making sure that each equation represents the same variable terms lined up vertically.
• Next, you aim to adjust the equations by multiplying them, if necessary, so that when you add or subtract them, one variable will cancel out.

The goal is to produce an equivalent system that is easier to solve. By focusing down on one variable, the system can be reduced to a single equation, making the process much simpler. Once one variable is solved, substitute back to find the other. This approach works smoothly unless the system is inconsistent or the equations represent the same line.

Inconsistent Systems

An inconsistent system is a system of equations that has no solution.
This occurs when the equations represent parallel lines that never intersect. It happens when both equations have the same slope but different y-intercepts.

• In mathematical terms, when the coefficients of the variables are proportional, but the constant terms are not, the system is inconsistent.
• If simplified, the equations may look like: \[ax + by = c \]and \[ax + by eq c \] This indicates that these equations form parallel lines.

An inconsistent system is identifiable during the solution process, whether you're using elimination, substitution, or graphical methods, because the resulting equations or lines show no points in common.

Parallel Lines

In the context of linear equations, parallel lines are lines in a plane that never meet.
They have the same slope, meaning they rise and run at the same rate, but different y-intercepts ensure they are distinct and never cross.

• When a system of equations results in parallel lines, the system is often inconsistent, since there are no intersection points.
• If you encounter a system where both equations simplify to the same slope but different y-intercepts, it’s a clear sign the lines are parallel.For example, equations like: \[ 9x - 5y = 1 \]and, \[ 9x - 5y = -\frac{1}{2} \], will form parallel lines.

Understanding parallel lines is not only crucial in solving systems of equations but also provides insights into the concepts of geometry and slopes.