Problem 38 - Algebra for College Students

Question

Solve each system by using the substitution method. \(\left(\begin{array}{l}7 x+2 y=11 \\ 7 x+2 y=-4\end{array}\right)\)

Step-by-Step Solution

Answer

The system has no solution because the lines are parallel.

Step 1: Identify the System of Equations

First, let's write down the given system of equations: \(7x + 2y = 11\) and \(7x + 2y = -4\). We need to solve this system using substitution.

Step 2: Recognize the System is Identical

Upon observing the two equations, notice they are parallel since both have the exact same left-hand side but different right-hand sides: \(7x + 2y = 11\) and \(7x + 2y = -4\). This means they represent two parallel lines with no points in common.

Step 3: Conclusion on No Solution

Since the two lines have the same coefficients for "x" and "y" but different constant terms, they never intersect. Thus, there are no solutions to this system, as parallel lines never meet.

Key Concepts

substitution method

The substitution method is a powerful tool for solving a system of linear equations. It involves solving one of the equations for one variable and then substituting that expression into the other equation. This technique can turn a system of equations into a single equation with one variable, making it much simpler to solve. For example, if you have two equations, like \(7x + 2y = 11\) and \(7x + 2y = -4\), the idea is to express either \(x\) or \(y\) from one equation, and then substitute it into the other equation.

In our specific scenario, however, both equations look identical on the left side, \(7x + 2y\), but have different right-hand sides (11 and -4). In this case, even before starting with substitution, we can see an anomaly that points out an important consequence, which leads us to parallel lines. It turns out substitution isn't needed once we recognize this pattern, saving time and effort!

parallel lines

Parallel lines are a fascinating concept, especially when dealing with linear equations in a coordinate plane. Two lines are parallel if they have identical slopes, meaning they rise and run at the same angles, but they never touch or intersect each other. The slope of a line in a linear equation \(ax + by = c\) is determined by the ratio \(-a/b\). When comparing two lines, if these ratios are the same, the lines are parallel.

In the given exercise, both equations \(7x + 2y = 11\) and \(7x + 2y = -4\) have the same coefficients for \(x\) and \(y\), resulting in the same slope. This identical slope confirms that the lines are parallel. Each line moves through the coordinate plane in the same direction but starts and remains apart due to different intercepts (11 and -4 in this case). Recognizing parallel lines helps us understand the nature of the solutions or the lack thereof in such a system.

no solution in systems of equations

A system of equations can have different types of solutions. It might have one solution, infinitely many solutions, or no solution at all. In cases where no solution exists, the equations represent parallel lines. These lines will never meet or intersect, hence there is no common point that satisfies both equations. This is exactly what happens in the exercise you are working on.

When you deal with \(7x + 2y = 11\) and \(7x + 2y = -4\), after recognizing they are parallel, it's clear there's no single \((x, y)\) pair that can satisfy both conditions simultaneously. Understanding that parallel lines mean no intersection is key to quickly identifying when a system of equations has no solution. Often, students begin to solve these using traditional methods before realizing the unique nature of the lines, so recognizing this early can simplify and hasten your work.