Problem 49 - 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. $$\left\\{\begin{array}{l}3 x+y=0 \\ y=-3 x+1\end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations has no solution, as the slopes of the two equations are the same, but the y-intercepts are different.

Step 1: Determine the Slope and y-Intercept for the First Equation

The first equation is \(3x + y = 0\), which can be rearranged into the slope-intercept form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. So, rearrange the equation to get \(y = -3x\). Thus, the slope of the first equation is -3, and the y-intercept is 0.

Step 2: Determine the Slope and y-Intercept for the Second Equation

The second equation is already in slope-intercept form (\(y = -3x + 1\)). So here, the slope is -3 and the y-intercept is 1.

Step 3: Compare the Slope and y-Intercept of the Two Equations

The slopes of the two equations are the same (-3), but their y-intercepts are different (0 and 1). Therefore, the system has no solution.

Key Concepts

Slope and y-intercept

Understanding the slope and y-intercept is fundamental when it comes to analyzing the behavior of linear equations. The slope, often represented as the letter 'm', describes how steep a line is. In simpler terms, it tells you how much the line goes up or down as you move along it, and is calculated as the 'rise over run' or the change in y divided by the change in x.

On the other hand, the y-intercept is where the line crosses the y-axis. This is typically represented by the letter 'c' or 'b' in the equation and tells us the value of y when x equals zero. In the context of the given exercise, the first equation gives us a slope of -3 and a y-intercept of 0, while the second equation presents the same slope of -3 but with a different y-intercept of 1.

Slope-intercept form

The slope-intercept form of a line is an equation of the form \( y = mx + c \). It is one of the most convenient ways of writing a linear equation because it directly shows both the slope 'm' and the y-intercept 'c'.

For example, let's consider the equations provided in the exercise. The first equation initially wasn't in the slope-intercept form, but with a bit of algebra, we rearranged it to \( y = -3x \), clearly displaying that the slope is -3 and the y-intercept is 0. The second equation was already presented in slope-intercept form (\( y = -3x + 1 \) ), making it easy to discern the same slope and a different y-intercept at a glance.

Algebraic solutions

Solving systems of equations often involves finding algebraic solutions which indicate where the graphs of the equations intersect. If the slopes are different, the lines will eventually cross, giving us a single solution representing the intersection point. When the slopes are identical but y-intercepts differ, as with our equations, the lines are parallel and will never meet, leading to no solution. A unique circumstance arises when both the slopes and y-intercepts are the same; the lines lay on top of each other, hence there are infinitely many solutions since they intersect at every point.

Graphical representation of equations

The graphical representation involves plotting lines on the Cartesian plane, which often provides a visual confirmation of what the algebra implies. In the case of the exercise, graphing the two equations would reveal that the lines are parallel and never intersect, visualizing the fact that there's no solution. Even without plotting, knowing that the lines have the same slope but different y-intercepts tells us enough to conclude they are parallel. Graphical methods can be handy when dealing with more complex systems, offering a visual alongside the algebraic approach.