Problem 4 - Algebra 1: Concepts and Skills

Question

Graph the system of linear equations. Does the system have exactly one solution, no solution, or infinitely many solutions? Explain. $$\begin{array}{c} {2 x+y=5} \\ {-6 x-3 y=-15} \end{array}$$

Step-by-Step Solution

Answer

The system of equations has infinitely many solutions since the lines are identical, meaning every point on one line lies on the other line.

Step 1: Reorganize the Equations

Begin by reorganizing both equations into slope-intercept form, \(y= mx + b\). For equation 1, \(2x+y=5\), isolate \(y\) to find \(y=-2x+5\). For equation 2, \(-6x-3y=-15\), isolate \(y\) to get \(y = 2x + 5\).

Step 2: Graphing the Equations

Next, graph these lines onto a Cartesian plane. Note that both lines will intersect at the y-axis value of 5. The first line will have a slope of -2, resulting in a downward trend, and the second line will have a slope of 2, resulting in an upward trend.

Step 3: Determine Number of Solutions

Upon graphing, you will notice that the lines are parallel and do not intersect. Since these lines are identical, every point on one of the lines also lies on the other line. Thus, the system of equations has an infinite number of solutions.

Key Concepts

Slope-Intercept Form

When dealing with linear equations, an important format to know is the slope-intercept form, which is defined as \( y = mx + b \).
Here, \( m \) represents the slope and \( b \) is the y-intercept, which is where the line crosses the y-axis.
This form makes it easy to both interpret and graph linear equations since it clearly shows the line's steepness and its starting point on the graph.

To convert a general linear equation to this form, all you need to do is solve for \( y \).
For example, given the equation \( 2x + y = 5 \), you can isolate \( y \) by subtracting \( 2x \) from both sides.
This transformation gives us \( y = -2x + 5 \), making it easy to see that the slope \( m \) is \(-2\) and the y-intercept \( b \) is \(5\).
This method of conversion represents a fundamental step in graphing linear equations and solving linear systems through visualization.

Graphing Linear Equations

Graphing linear equations provides a visual way to comprehend and analyze solutions.
Once in slope-intercept form \( y = mx + b \), graphing is straightforward.

You start by locating the y-intercept \( b \) on the y-axis.
For instance, in the equation \( y = -2x + 5 \), you start at point \( (0, 5) \) on the y-axis.
Next, use the slope \( m \), typically represented as a fraction for convenience, \( m = -2/1 \), meaning we go two units down and one unit right from the y-intercept to plot another point.

By drawing a straight line through these points, we graph the equation.
For multiple linear equations, plotting them on the same graph helps determine intersections, which represent common solutions.
Parallel lines, when graphed, show inconsistent systems with no solutions. Identical lines, however, overlay each other, indicating potentially infinite solutions.

Infinite Solutions

In systems of linear equations, infinite solutions occur when the lines in the system perfectly overlap each other on the graph.
This means any point on one line is also on the other line, indicating the equations are essentially the same after simplification.

Such systems are described as dependent because one equation can be derived from the other by algebraic manipulation.
For example, if you take both equations from the original exercise, the equations \( 2x + y = 5 \) and \(-6x -3y = -15 \), converting them to slope-intercept form reveals \( y = -2x + 5 \) and \( y = 2x + 5 \), respectively.
Comparing these, you can see they represent the same line equation.
Graphically, this is visible when both equations situate on top of each other, thus every point on the line is a solution.

So, understanding this concept is crucial when manipulating and interpreting results, especially when equations initially appear distinct but reveal identical relationships through simplification.