Problem 45 - 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. $$\left\\{\begin{array}{l}y=-\frac{1}{2} x+4 \\ 3 x-y=-4\end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations has one solution.

Step 1: Identify Slope and Y-Intercept for each Equation

For the first equation \(y=-\frac{1}{2}x+4\), the slope \(m_1\) is -1/2 and the y-intercept \(c_1\) is 4. For the second equation, it needs to be rewritten in slope-intercept form. So, rewriting \(3x - y = -4\) gives \(y = 3x +4\), here the slope \(m_2\) is 3 and the y-intercept \(c_2\) is 4.

Step 2: Compare the Slopes and Y-Intercepts

The slopes \(m_1\) and \(m_2\) of the two equations are different (-1/2 and 3 respectively) but the y-intercepts \(c_1\) and \(c_2\) are the same (4). This indicates that the lines represented by the equations cross the y-axis at the same point but are not parallel.

Step 3: Determine the Number of Solutions

Since the slopes are not equal, the lines will intersect at certain x and y values, meaning the system of equations has one solution.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most widely used ways to express a linear equation in algebra. It allows us to easily identify two critical components of a line: the slope and the y-intercept.

• The slope, usually represented by the letter \(m\), shows the steepness or incline of a line. It indicates how much the line goes up or down for every horizontal unit moved.
• The y-intercept, on the other hand, symbolizes the point where the line crosses the y-axis. It's denoted by \(b\) in the equation \(y = mx + b\).

Putting an equation into this form helps you quickly determine the slope and y-intercept by simply looking at the numbers. For instance, in the equation \(y = -\frac{1}{2}x + 4\), the slope \(m\) is \(-\frac{1}{2}\) and the y-intercept is 4, which tells us how the line positions itself on a graph.

Y-Intercept

The y-intercept is particularly vital because it tells you exactly where the line crosses the y-axis, making it easy to plot the first point on a graph. When an equation is written in slope-intercept form, this value is straightforwardly represented by \(b\).
In our given equations, both have a y-intercept of 4, which means that both lines will cross the y-axis at the same vertical position. This piece of information is useful when determining the nature of the lines involved since knowing where the lines intersect the y-axis helps to understand their relative positions and potential intersections with each other.
Visualizing that both lines meet at the y-axis in the same spot but have different slopes immediately tells us about their orientation and how they might intersect elsewhere on the graph.

Solving Equations

Solving a system of equations involves finding values for the variables that satisfy all given equations simultaneously. When working with linear systems, we're basically looking for the point(s) where the lines represented by the equations intersect.
The primary tools in this process are the slopes and y-intercepts of the lines, which we have already found. Once we know these, we can:-

• Compare the slopes to see if the lines are parallel, intersect at one point, or coincide entirely, leading to either no solution, one solution, or infinite solutions, respectively.


• In our current problem, the different slopes of \(-\frac{1}{2}\) and 3 show us that the lines indeed intersect at exactly one point.

This means the system has one solution—specifically, the point where these two lines cross each other on the graph. Having a clear understanding of this allows us to solve these systems methodically and verify the number of solutions confident that our results are accurate.