Problem 27 - Algebra 1

Question

Use the graphing method to solve the linear system and tell how many solutions the system has. $$ \begin{aligned}&-x+4 y=-20\\\&3 x-12 y=48\end{aligned} $$

Step-by-Step Solution

Answer

The given system of equations has no solutions as the lines representing the equations are parallel.

Step 1: Rewrite the equations into slope-intercept form

The equation in slope-intercept form is defined by y = mx + c where m is the slope and c is the y-intercept. Thus, the given equations yield:\\( y = 0.25x + 5 \\)\\( y = 0.25x + 4 \\)

Step 2: Comparing Equations

Looking at both equations, we can see that both lines have the same slope (0.25), but different y-intercepts (5 and 4). This implies that we have two parallel lines.

Step 3: Analyzing the graph

Parallel lines never intersect each other. This means that there are no points that lie on both lines at the same time. Hence, the system has no solution.

Key Concepts

Graphing Method

The graphing method is a visual way to find the solution to a system of equations. It involves plotting two lines on a coordinate plane to see where they intersect. When applying the graphing method, you first need to rewrite the equations in a form that's easy to graph, typically the slope-intercept form, which is expressed as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. After graphing both lines, the point of intersection represents the solution to the system. However, if the lines do not cross and are parallel, as is the case in our example, there will be no solution to the system. To ensure accurate graphing, always plot more than one point for each line and use a ruler to draw straight lines.

Slope-Intercept Form

The slope-intercept form is the equation of a line written as \( y = mx + b \), where \( m \) and \( b \) are constants. This form is incredibly useful because it directly tells you the slope, \( m \), which indicates the steepness and direction of the line, and the y-intercept, \( b \), which is the point where the line crosses the y-axis. When graphing, start by plotting the y-intercept on the vertical axis. Then, use the slope as a fraction to determine the rise over run, helping to find the next points. For example, if the slope is \( \frac{1}{2} \), go up 1 unit and right 2 units to plot the next point. It's crucial to understand this form to effectively graph linear systems.

Parallel Lines

Parallel lines are straight lines in the same plane that never intersect each other, no matter how far they are extended. In the context of linear equations, two lines are parallel if and only if they have the same slope but different y-intercepts. Recall the slope-intercept form \( y = mx + b \): the value of \( m \) determines the slope. When we have two equations with identical slopes but different y-intercepts, this visually translates to two parallel lines. Since they do not meet, these lines do not share any common points, implying there is no set of \( x \) and \( y \) values that satisfy both original equations simultaneously.

No Solution Systems

In algebra, a system of equations may be categorized by the number of solutions it has: one, infinitely many, or none at all. A no solution system, also known as an inconsistent system, occurs when equations represent parallel lines. Since these lines never touch, there is no point that satisfies both equations, leading to the conclusion there is no solution. Recognizing a no solution system early on can save you time. By converting equations to slope-intercept form and observing the slopes and y-intercepts, you can determine if you are dealing with parallel lines, hence a system with no solution, without necessarily having to graph the equations.