Problem 43 - 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. $$\left\\{\begin{array}{l}y=\frac{1}{2} x-3 \\ y=\frac{1}{2} x-5\end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations has no solution.

Step 1: Identify the slope and y-intercept of each equation

For the first equation \(y = \frac{1}{2}x - 3\), the slope \(m1 = \frac{1}{2}\) and the y-intercept \(b1 = -3\).\nFor the second equation \(y = \frac{1}{2}x - 5\), the slope \(m2 = \frac{1}{2}\) and the y-intercept \(b2 = -5\).

Step 2: Compare the Slopes and Y-intercepts

The slopes of both equations are equal, \(m1 = m2\), but the y-intercepts are not (i.e., \(b1 \neq b2\)).

Step 3: Determining the nature of solutions

According to the rules of system of equations, when the slopes of the two equations are same and the y-intercepts are different, the lines are parallel to each other and thus they will never intersect. This means the system has no solution.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial when analyzing linear equations. The slope is a measure of how steep a line is. In mathematical terms, it represents the ratio of the rise (the vertical change) to the run (the horizontal change) between two points on a line. The slope is often represented by the letter 'm'.

If you have an equation in the slope-intercept form, which is written as \(y = mx + b\), the coefficient of \(x\) is your slope. When comparing slopes, if two lines have the same slope, they are either parallel or the same line. In our exercise, both lines have a slope of \(\frac{1}{2}\), indicating that these lines are parallel or identical. Since the y-intercepts differ, we can conclude they are parallel and will never cross each other.

Y-intercept

Moving onto the y-intercept, this is the point where the line crosses the y-axis on a graph. It is where the value of x is zero. The y-intercept is represented by the 'b' in the slope-intercept form \(y = mx + b\).

Knowing the y-intercept is useful for graphing the line and understanding its position relative to the origin. Each line in our exercise equation has a different y-intercept; the first has a y-intercept of -3 and the second -5. This difference in y-intercepts, while having identical slopes, ensures that the lines do not intersect, as they are offset vertically from one another.

Parallel Lines

Parallel lines are straight lines in a plane that never meet; they are always the same distance apart and have the same slope. The characteristic of parallel lines is essential in our exercise as it directly determines the nature of the solutions for the system of equations.

Since the slopes of the given equations are identical, and the y-intercepts are different, we can deduce that the lines will never intersect. This is because one line is a 'shift' above or below the other, but they are moving in the same direction because of the identical slope.

Number of Solutions

The number of solutions in a system of linear equations can vary. If two lines intersect at a single point, there's one unique solution (that point of intersection is the solution). If they are identical (coincide completely), there is an infinite number of solutions as every point on the line satisfies both equations.

However, in our textbook exercise, since the lines are parallel (same slope, different y-intercept) they will never cross. As a result, there is no point that lies on both lines at the same time, which means the system has no solution. It's essential to recognize the relationship between the slopes and y-intercepts of equations within a system to determine the number of solutions.