Problem 46 - 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. $$\left\\{\begin{array}{l}y=-\frac{1}{4} x+3 \\ 4 x-y=-3\end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations has one solution because the lines they represent intersect at one point, which is indicated by their different slopes and y-intercepts. The slope of the first equation is -1/4, the y-intercept is 3, the slope of the second equation is 4 and the y-intercept is also 3.

Step 1: Convert Equation to Standard Form

The first equation is already represented in the slope-intercept form \(y = mx + b\), but we need to put the second equation into an equivalent form, specifically into slope-intercept form. This is done by rearranging the terms so that we get \(y = 4x + 3\)

Step 2: Find the Slopes for both Equations

Comparing the equations to the form \(y = mx + b\), we can see that the slope of the first equation and the slope of the second equation both are -1/4 and 4 respectively.

Step 3: Find the Y-intercepts for both Equations

We can look at the equations \(y = -1/4x + 3\) and \(y = 4x + 3\) to find out that the y-intercepts for the first and second equations are both 3. This is because the y-intercept is represented by \(b\) in the equation \(y = mx + b\).

Step 4: Compare the Slopes and Y-intercepts

The two lines represented by these equations have different slopes (-1/4 and 4), and different y-intercepts (3 and 3). Therefore, we can conclude that the lines represented by these equations will intersect at exactly one point. That means there is one solution for this system of equations.

Key Concepts

Understanding Slope

The slope is a crucial concept in understanding the nature of a line in coordinate geometry. The slope of a line indicates its steepness and is represented by the letter \(m\) in the slope-intercept form of an equation, \(y = mx + b\). The value of the slope tells us how much \(y\) changes as \(x\) changes by 1 unit. - A positive slope means that the line rises as it moves from left to right. - A negative slope means that the line falls as it moves from left to right. - A slope of zero indicates a horizontal line, which means no vertical change.
In our exercise, the first equation \(y = -\frac{1}{4}x + 3\) has a slope of \(-\frac{1}{4}\). This means that for every 4 units the line moves horizontally to the right, it moves 1 unit downward. The second equation \(y = 4x + 3\) has a slope of 4, indicating the line moves upward, 4 units for every 1 unit it moves horizontally to the right.

What is the Y-intercept?

The y-intercept is where the line crosses the y-axis. It is the value of \(y\) when \(x\) is zero and is denoted by \(b\) in the equation \(y = mx + b\). - The y-intercept gives us a starting point for drawing the line. - It's a valuable piece of information when graphing the equation, as it tells us exactly where the line meets the y-axis.
For the equations in our problem, both lines cross the y-axis at the same point, \(y = 3\). Even though the lines have different slopes, they start from the same y-intercept.

Solving Systems of Equations

A system of equations is a set of two or more equations with the same variables. The objective is to find values for those variables that satisfy all equations simultaneously. There are a few key possibilities: - **No solution:** Lines are parallel and never intersect. - **One solution:** Lines intersect at exactly one point. - **Infinite solutions:** Lines are identical.
In the given exercise, the lines are not parallel because they have different slopes (-1/4 and 4). This means they will intersect. Since their y-intercepts are equal, yet slopes are different, they will intersect just once, thus providing precisely one solution.

Slope-Intercept Form Explained

The slope-intercept form, \(y = mx + b\), is a method to write linear equations in an easy-to-read format. - **\(m\) is the slope:** This tells us how the line moves across the graph. - **\(b\) is the y-intercept:** This tells us the starting point of the line on the y-axis.
Why is this useful? - **Simplicity:** It gives a clear picture of the line's orientation (upward/downward) and its crossing point on the y-axis. - **Graphing ease:** When you know the slope and y-intercept, you can quickly sketch or understand the line's behavior without needing a graph.
In our exercise, adapting both equations into slope-intercept form allowed us to straightforwardly compare slopes and y-intercepts. This reveals the nature of their intersection, better helping us understand and solve the system.