Problem 37 - Precalculus: Mathematics for Calculus

Question

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. $$\left\\{\begin{array}{r} x+4 y=8 \\ 3 x+12 y=2 \end{array}\right.$$

Step-by-Step Solution

Answer

The system has no solution because it represents parallel lines.

Step 1: Check the consistency of the system

To determine if the system is consistent, rewrite both equations in standard form and observe if they represent parallel lines. The given equations are: \(x + 4y = 8\) and \(3x + 12y = 2\). Rewrite the second equation by dividing every term by 3: \(x + 4y = \frac{2}{3}\).

Step 2: Compare equations

Notice that the first equation is \(x + 4y = 8\) while the second is \(x + 4y = \frac{2}{3}\). Since the left-hand sides (the coefficients of \(x\) and \(y\)) of both equations are identical but the right-hand sides are different, these equations represent parallel lines that never intersect.

Step 3: Conclude that the system has no solution

Parallel lines do not intersect, hence the system is inconsistent and has no solutions. This means that the set of equations has no pair \((x, y)\) that satisfies both equations simultaneously.

Key Concepts

Inconsistent Systems

An inconsistent system of linear equations is one where no solution exists. In other words, there is no set of values for the variables that will satisfy all equations simultaneously. This usually happens when the equation lines are parallel, yet their intercepts on the y-axis differ. To break it down, you are essentially looking at lines that never touch—hence, they cannot share any point in common. The system mentioned in the exercise is a classic case of this inconsistency; both lines have the same slope but different y-intercepts. By observing this characteristic, we can conclude that the system is indeed inconsistent.

Parallel Lines

Parallel lines occur when two or more lines have the same slope but different y-intercepts. Imagine they lie flat on a plane, moving in the same direction but never meeting. This is because, mathematically, parallel lines have identical coefficients for their variables when rewritten into standard line equations of the form:

• Standard Form: \( ax + by = c \)

In the given solution, the equation forms \( x + 4y = 8 \) and \( x + 4y = \frac{2}{3} \) are both normalized to show they parallel each other. Their slopes are essentially \( -\frac{1}{4} \), maintaining independence in their paths without crossing. Thus, they never intersect, confirming the lines are parallel.

Solving Linear Equations

Solving linear equations involves finding the values that satisfy each equation in a system. When working with two-variable linear equations, you usually find the point where their graphs intersect, known as their solution. The primary methods include substitution, elimination, and graphing. However, recognizing line relationships such as parallelism can save time. If you attain similar equations with differing constants, you instantly note every possible step points to an inconsistent system.

• Substitution Method: Replace one variable with its equivalent from another equation.
• Elimination Method: Add or subtract equations to eliminate a variable, simplifying the solution.
• Graphing: Plot each equation to visualize intersection points.

In this exercise, constructing a visual graph is unnecessary due to the realization that there is nothing to solve. The parallel characteristic flags this instantly as having no common solution.