Question

Use the graphing method to solve the linear system and tell how many solutions the system has. $$ \begin{aligned}&3 x-2 y=3\\\&-6 x+4 y=-6\end{aligned} $$

Short answer

The system has infinitely many solutions as the lines representing the equations coincide in the graphical representation.

Step-by-step solution

Write the equations in Slope-Intercept Form

Rewrite both equations in the form \(y = mx + b\), so it's easier to graph. For the first equation, add \(2y\) to both sides and divide by 2, you get \(y = (3/2)x - 3/2\). The second equation simplifies to \(y = (3/2)x - 3/2\).

Graph the System of Equations

Graph both equations on the same graph. Since they are the same equation, the lines representing them will coincide.

Analyze the Graph

Looking at the graph, it is clear that the lines coincide. This means that every point on the line is an intersection point, i.e. it satisfies both equations.

Determine the number of solutions

A system of equations that coincides in their graphical representation means that they share all their solutions, thus the system has infinitely many solutions.

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial for graphing linear systems. It is represented as ( y = mx + b ), where m represents the slope and b signifies the y-intercept, the point where the line crosses the y-axis. The slope determines how steep the line is and whether it ascends or descends as it moves from left to right. Positive values of m mean an uphill slope, while negative values indicate a downhill slope.

To put an equation into slope-intercept form, we often need to solve for y in terms of x. This could include moving terms from one side of the equation to the other and dividing by the coefficient of y. The exercise demonstrates this by transforming (3x - 2y = 3) and (-6x + 4y = -6) into slope-intercept form, resulting in (y = (3/2)x - 3/2) for both. This particular form makes graphing straightforward, as you can identify the starting point on the y-axis and the direction and steepness of the line based on the slope.

Visualizing and graphing

When dealing with graphing, it is important to choose suitable scale and intervals to depict the line accurately on a graph. Students can plot the y-intercept first and then use the slope as a guide to determine other points on the line. This step-by-step approach simplifies the graphing process and ensures that students can easily graph any linear equation.

System of Equations

A system of equations is a set of equations with multiple variables that you solve at the same time. The objective is to find a common solution that satisfies all equations simultaneously. When graphing linear systems, you're looking for points where the lines intersect, representing the solutions. The system can have one solution (intersecting lines), no solution (parallel lines), or infinite solutions (coincident lines).

In this exercise, students are tasked with graphing two linear equations to find their point or points of intersection. By following the step-by-step solution, students learn that if the equations are manipulated into the same slope-intercept form, implying the lines have identical slopes and y-intercepts, they will graph as the same line. This is a fundamental insight into recognizing the special situation where systems have infinite solutions.

Common pitfalls

Students must be cautious to properly simplify and rearrange the equations into slope-intercept form without making algebraic errors, as incorrect manipulation could lead to a misunderstanding of the system's solutions. Additionally, it's essential to graph with precision to identify the nature of the intersection accurately.

Infinite Solutions

The concept of infinite solutions in the context of a linear system of equations indicates that the equations describe the same line. This happens when the slope and y-intercept of both equations are identical after being transformed into slope-intercept form. As seen in the exercise, both equations simplified to (y = (3/2)x - 3/2), signifying that every point on the line satisfies both equations, hence an infinite number of solutions.

It's important to identify this scenario as it has implications across mathematics and its applications. The existence of infinite solutions means that the two equations provide the same information and are not independent. Students may encounter this when dealing with dependent systems in algebraic content or real-world problems requiring system analysis.

Understanding the implications

In practical terms, finding infinite solutions can inform us that we lack a unique solution and may need more information to find a distinct answer to a problem. In the context of graphing, careful analysis of the graph helps conclude the nature of the solutions, which has been clearly modeled through the steps provided in the exercise.