Problem 21 - Algebra and Trigonometry

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

Step-by-Step Solution

Answer

The system has no solution as the lines are parallel with different intercepts.

Step 1: Analyze the system of equations

The given system of equations is: \( x + 4y = 8 \) and \( 3x + 12y = 2 \). Note that the second equation seems to be a multiple of the first, so let's simplify it to determine the relationship.

Step 2: Simplify the second equation

Divide the entire second equation by 3: \[ \frac{3x + 12y}{3} = \frac{2}{3} \]. This simplifies to \( x + 4y = \frac{2}{3} \).

Step 3: Compare the simplified equations

Compare the simplified second equation \( x + 4y = \frac{2}{3} \) with the first equation \( x + 4y = 8 \). The coefficients are identical, but the constants on the right side are different, \( \frac{2}{3} eq 8 \).

Step 4: Conclude about the system

Since the equations lead to the same linear combination of variables but different constant terms, they are parallel lines and never intersect. Thus, the system has no solution.

Key Concepts

Parallel Lines

In a system of linear equations, each equation represents a line in the coordinate plane. The concept of parallel lines arises when these lines have the same slope. For equations in the form of \( ax + by = c \), they are parallel if the ratios \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) hold true, but \( \frac{c_1}{c_2} \) does not. This means the lines have identical directions but will never meet.

In the given system, the equations \( x + 4y = 8 \) and \( x + 4y = \frac{2}{3} \) have equal coefficients of \( x \) and \( y \). Both equations simplify to the same slope, confirming that they are parallel. However, their different constants mean these lines are parallel but distinct, leading to no point of intersection.

No Solution

When equations in a system depict parallel lines, as in this case, the system typically has no solution. This happens because parallel lines never intersect. Each line remains separate forever, never crossing one another.

For a system of equations, finding no solutions means looking for a consistent pattern:

• Equations share equal linear coefficients but different constants.
• They simplify to parallel lines.
• The system yields incompatible conditions, making it unsolvable.

By inspecting these factors, you can quickly determine when a system lacks solutions, saving time and effort that could be wasted on further calculations.

Linear Equation Simplification

Simplifying equations is a vital skill in solving systems. It involves manipulating an equation to reveal its essential properties. This might include dividing, adding, or subtracting terms.

In the initial problem, simplifying was key. For the equation \( 3x + 12y = 2 \), dividing every term by 3 provided \( x + 4y = \frac{2}{3} \). Simplifying clarifies whether the equations represent the same line, parallel lines, or something else.

• It helps identify key characteristics of the equations.
• It provides a more straightforward path to conclusions.
• Simplified forms are easier to compare, enhancing understanding.

Engaging in simplification is often the first insight into the structural relationship of the equations, enabling you to solve or analyze the system with clarity.