Problem 50 - Introductory Algebra for College Students

Question

Find the slope and the \(y\) -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions. $$\left\\{\begin{array}{l}2 x+y=0 \\ y=-2 x+1\end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations has no solution because the lines represented by the equations are parallel (having the same slope but different y-intercepts).

Step 1: Re-arrange the first equation to the form \(y = mx + c\)

The given equation is \(2x + y = 0\). We can re-arrange that to the form \(y = -2x\). So, the slope of the line is \(-2\) and the y-intercept is \(0\).

Step 2: Identify slope and y-intercept for the second equation

The second equation is already in the form \(y = mx + c\), i.e, \(y = -2x + 1\). Its slope is \(-2\) and y-intercept is \(1\).

Step 3: Compare the slopes and y-intercepts of the two equations

Both equations have the same slope \(-2\), but their y-intercepts are different. \(0\) for the first equation and \(1\) for the second equation. This means that the two lines are parallel and do not intersect.

Key Concepts

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in the form of y = mx + b, where y represents the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept. The slope is a measure of how steep a line is, and the y-intercept is the point where the line crosses the y-axis.

The slope m is determined by the change in y over the change in x between two distinct points on the line. If the slope is positive, the line slants upward as we move from left to right. Alternatively, if the slope is negative, the line slants downward.

In the provided exercise, the equations given are part of a system and need to be rearranged to identify the slope and y-intercept. For example, converting 2x + y = 0 to y = -2x reveals a slope of -2 and a y-intercept of 0. Understanding the standard form of linear equations is essential as it allows us to analyze and compare their characteristics, such as the slope and y-intercept.

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. Systems of linear equations can have one solution, no solution, or infinitely many solutions. When two or more equations are graphed, their solution is the point or points where the lines intersect.

A unique solution occurs when there is exactly one point that satisfies all equations in the system. This happens when the lines intersect at a single point, indicating the lines are not parallel and have different slopes.

When there is no solution to the system, the lines are parallel and don't cross each other at any point. In our exercise above, the fact that the two lines have identical slopes but different y-intercepts (-2 for both slopes, 0 and 1 for y-intercepts) tells us that the lines will never intersect, thus, there's no solution to this system.

An infinite number of solutions means the lines are coincident, that is, they lie on top of each other, indicating that the equations are just different representations of the same line and therefore have both the same slope and the same y-intercept.

Parallel Lines

Parallel lines are two or more lines on a plane that never meet; they remain the same distance apart over their entire length. In terms of linear equations, parallel lines have identical slopes but different y-intercepts.

In the context of the exercise we are discussing, recognizing parallel lines through their slopes is crucial. Since both equations 2x + y = 0 and y = -2x + 1 have the slope -2, we can conclude that these lines are parallel. The fact that their y-intercepts do not match (0 for the first line and 1 for the second) reinforces this concept as it confirms that these lines will never cross.

Understanding the characteristics of parallel lines is valuable when solving systems of equations. It helps us quickly determine when a system of linear equations has no solution. Remember, if you change a y-intercept of a line, you're simply sliding it up or down the y-axis, without affecting its slope, and thus its parallelness to other lines remains unchanged.