Problem 47 - Algebra 2

Question

Solve each system of equations by graphing. $$ \begin{array}{l}{2 x+y=-3} \\ {6 x+3 y=-9}\end{array} $$

Step-by-Step Solution

Answer

Infinitely many solutions; both equations represent the same line.

Step 1: Convert to Slope-Intercept Form

Begin by rewriting each equation in the slope-intercept form, which is given as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For the first equation, \( 2x + y = -3 \), solve for \( y \) to get \( y = -2x - 3 \). For the second equation, \( 6x + 3y = -9 \), divide each term by 3 to get \( 2x + y = -3 \). Notice both equations simplify to the same line \( y = -2x - 3 \).

Step 2: Recognize System Properties

Both equations represent the same line \( y = -2x - 3 \). This means they are dependent and consistent, meaning there are infinitely many solutions since both represent the same line.

Step 3: Graph the Lines

Graph the line \( y = -2x - 3 \). Start at the y-intercept \((0, -3)\) and use the slope \(-2\) to find another point, such as \((1, -5)\), obtained by moving 1 unit to the right (increase in \(x\)) and 2 units down (decrease in \(y\)). Both equations will graph to this same line.

Step 4: Determine the Solution

Since the two equations graph to the same line, any point on this line is a solution to the system. Therefore, there are infinitely many solutions.

Key Concepts

Graphing Systems of Equations

Graphing systems of equations is a visual method to find the solutions where two or more equations intersect on a graph. When solving a system by graphing, you are essentially looking for the point(s) where the graphs of the equations meet. These points are the solutions to the system.
For linear equations in two variables, this involves graphing lines on the coordinate plane:

• Each equation is represented as a line.
• The point(s) of intersection are the solutions.

Start by converting each equation into a graphable form, like the slope-intercept form, which helps make plotting straightforward. After graphing, observe how the lines interact:

• If they intersect at a single point: The system has a unique solution.
• If they are parallel: The system has no solution.
• If they are the same line: The system has infinitely many solutions.

Slope-Intercept Form

The slope-intercept form of a linear equation is written as \( y = mx + b \). This form allows for easy graphing and insight into the relationship described by the equation.
Here:

• \( m \) is the slope. It represents the rate of change or the steepness of the line. It tells you how much \( y \) changes for a unit change in \( x \).
• \( b \) is the y-intercept. This is the point where the line crosses the y-axis (when \( x = 0 \)).

To convert an equation into the slope-intercept form, solve for \( y \). This was used in the exercise to rewrite the given equations into \( y = -2x - 3 \). Notice how this form made it simple to analyze and graph the equations.

Dependent and Consistent System

A system of equations is described as dependent and consistent when its equations represent the same line. This means they have all their solutions in common, effectively having infinite solutions.
In simpler terms:

• Dependent: One equation is a multiple or transformation of the other.
• Consistent: There is at least one solution. In this case, every point on the line is a solution.

In the exercise example, both equations were transformed to \( y = -2x - 3 \), meaning they describe the same line. This indicates a dependent and consistent system, as they coincide completely and infinitely.

Infinite Solutions

Infinite solutions occur in a system of equations when there are countless solution points that satisfy each equation in the system. This is predominantly seen with dependent systems where the graphs of the equations coincide as one.

• When graphed, the lines overlap completely, appearing as a single line.
• Each point on the line shared by the equations is a solution.

This was seen in the example solution, where both equations simplified to \( y = -2x - 3 \). Therefore, any coordinate that lies on the line \( y = -2x - 3 \) is a valid solution to the system, displaying infinite solutions.