Problem 86 - College Algebra with Modeling and Visualization

Question

Solve the system, if possible. $$ \begin{array}{r} 5 x-2 y=7 \\ 10 x-4 y=6 \end{array} $$

Step-by-Step Solution

Answer

The system has no solution; the equations represent parallel lines.

Step 1: Write the System of Equations

The system of equations is given as follows: 1. \(5x - 2y = 7\)2. \(10x - 4y = 6\) We need to determine whether there is a solution, and if so, find it.

Step 2: Analyze the System for Multiplicities

Notice that the second equation \(10x - 4y = 6\) is exactly twice the first equation in terms of the coefficients of \(x\) and \(y\), but not for the constant term. This suggests that the two equations could be parallel lines but not the same line.

Step 3: Check for Consistency Using Substitution or Elimination

Using elimination, let's multiply the first equation by 2 to match the coefficients of the second equation:\[ 2(5x - 2y) = 2(7) \]\[ 10x - 4y = 14 \]Now compare it with the second equation \(10x - 4y = 6\). The left-hand side is identical, but the right-hand sides differ, indicating inconsistency.

Step 4: Determine the Solution Status

Since both equations have the left side \(10x - 4y\) equal but different right sides (14 and 6), this indicates that the lines represented by the equations are parallel and do not intersect. Hence, there is no solution.

Key Concepts

Equations

When dealing with linear systems, understanding equations is essential. In a system of equations, each equation represents a line on a plane. An equation like \(5x - 2y = 7\) is a linear equation in two variables, \(x\) and \(y\). Each point \((x, y)\) that satisfies this equation lies on a specified line. Similarly, another equation \(10x - 4y = 6\) also defines a line on the same plane. In solving a system of equations, one seeks points \((x, y)\) that satisfy all equations in the system simultaneously. These are known as solutions to the system. However, the system might have one solution, no solution, or infinitely many solutions depending on the nature of the lines involved.

Inconsistency

Inconsistency in a system of equations occurs when there are no common solutions that satisfy all equations. This often means the lines represented by these equations do not intersect anywhere on the graph. To check for inconsistency, you can use methods like substitution or elimination. In the problem provided, after analyzing the system of equations:- Multiply the first equation by 2 to align with the second equation: - \(2(5x - 2y) = 2(7)\) resulting in \(10x - 4y = 14\).- Compare this with the second equation: \(10x - 4y = 6\).The coefficients for \(x\) and \(y\) on both sides are identical, but the constants (14 and 6) differ. This difference signals inconsistency, implying no solution exists for the system as the equations describe parallel lines that never meet.

Parallel Lines

Parallel lines are lines in a plane that never meet; they remain the same distance apart over their entire length. They have identical slopes. To identify parallel lines in a system of equations like:- \(5x - 2y = 7\) and- \(10x - 4y = 6\),observe the ratio of the coefficients of both \(x\) and \(y\). When these ratios match, the lines are parallel unless the constant terms also match, in which case they would be the same line.- Here: - The coefficient ratio of the first equation \(5x - 2y\) is maintained in \(10x - 4y\), but differing constants (7 vs. 6 after multiplying) confirm that the lines are parallel.Since parallel lines never intersect, this feature explains why such a system is inconsistent, resulting in no solution.