Problem 38 - Algebra and Trigonometry

Question

\(21-48=\) Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example \(6 .\) $$ \left\\{\begin{array}{c}{2 x-3 y=-8} \\ {14 x-21 y=3}\end{array}\right. $$

Step-by-Step Solution

Answer

The system has no solution as the lines are parallel.

Step 1: Simplify the Second Equation

First, simplify the second equation \(14x - 21y = 3\) by dividing the entire equation by 7, which is the greatest common divisor of the coefficients.\[ 14x - 21y = 3 \implies 2x - 3y = \frac{3}{7}\]

Step 2: Compare the Equations

Now, compare the simplified form of the second equation \(2x - 3y = \frac{3}{7}\) with the first equation \(2x - 3y = -8\). Both have the same left-hand side but different right-hand side constants. This indicates that the two lines represented by these equations are parallel.

Step 3: Determine the Solution

Since the lines represented by equations \(2x - 3y = -8\) and \(2x - 3y = \frac{3}{7}\) have the same slope but different y-intercepts, they are parallel and do not intersect anywhere in the plane.

Key Concepts

Parallel Lines

Parallel lines are an important concept in the study of systems of linear equations. In simple terms, parallel lines are two lines that never intersect, no matter how far they are extended on the plane.
This happens because they have the same slope. The slope is a measure of how steep a line is, typically represented by the letter \(m\) in the equation of a line in slope-intercept form \(y = mx + b\).
When two lines have the same slope, they will rise and run at the same rate, ensuring they stay equidistant forever.

• If two lines are parallel, their equations will have identical left-hand sides when the equations are written in form \(Ax + By = C\).
• However, their right-hand sides will differ, resulting in different y-intercepts which causes them not to meet.

In the given problem, both equations have the same left-hand side \(2x - 3y\) but different constants on the right-hand side, leading to parallel lines.

No Solution

When dealing with systems of equations, one potential outcome is that the system has 'no solution.' This means there are no values for the variables that satisfy all equations simultaneously.
For a system of two linear equations, the 'no solution' condition explicitly means that the lines are parallel.
So, when you try to solve the system, you'll notice the equations indicate two lines with the same slope but different intercepts.

• Simplifying or rearranging the equations usually reveals the parallel nature.
• Because parallel lines do not intersect, there is no point \((x, y)\) that lies on both lines simultaneously.

In this scenario, observing that the equations \(2x - 3y = -8\) and \(2x - 3y = \frac{3}{7}\) don't intersect indicates this absence of a solution.

Linear Equations

A linear equation is an equation that forms a straight line when graphed. It typically takes the form \(Ax + By = C\), where \(x\) and \(y\) are variables.
Linear equations represent a constant relationship between these variables. This means that for every change in \(x\), there is a consistent change in \(y\).
Understanding linear equations is foundational for recognizing various relationships between quantities in math.

• A system of two linear equations can be solved by finding the point where the two lines intersect, if they do.
• If two linear equations describe the same line, they have infinitely many solutions, whereas if they are parallel as in this problem, they have none.

Linear equations are a core part of algebra and are critical for higher-level math studies and practical applications, such as economics and engineering.