Problem 18 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The system has an infinite number of solutions.

Step 1: Convert Equations into Slope-Intercept Form

Written in slope-intercept form, the first equation is \(y = \frac{3}{2}x - \frac{3}{2}\) and the second equation is \(y = \frac{3}{2}x - \frac{3}{2}\). Both equations have the same slope and intercept.

Step 2: Graph the Equations

Plot both equations on a graph. You will find that they overlap entirely because their slope and intercept are identical. Therefore, they are the same line or coincident lines.

Step 3: Count the Number of Intersections

Because the two lines coincide, it means they intersect at every point on the line. Therefore, the system has an infinite number of solutions.

Key Concepts

System of Linear Equations

A system of linear equations is a set of two or more linear equations that have common variables. These equations represent straight lines when graphed on a coordinate plane. The solution to a system is any point or set of points that satisfy all the equations in the system simultaneously.

There are three possible outcomes when solving a system of linear equations:

• One solution: The lines intersect at exactly one point.
• No solution: The lines are parallel and never intersect.
• Infinite solutions: The lines are identical and overlap completely.

Understanding how to solve these systems is crucial in fields like engineering and economics, where such equations often model real-world scenarios. In our exercise, converting equations into slope-intercept form helps easily identify the relationship between the lines.

Infinite Solutions

A system of linear equations has infinite solutions when the equations describe the same line. This means each equation is merely a different expression of the same line. Because they coincide perfectly, they intersect at every point along that line, resulting in infinite solutions.

In the exercise provided, after converting the equations into slope-intercept form, we see they have the same slope and intercept, indicating they are the same line.

When graphing these types of systems, the equations will perfectly overlap. This characteristic makes it straightforward to identify infinite solutions visually:

• The graphs are coincident lines.
• The slope is identical for both equations.
• The y-intercept is also the same.

Recognizing infinite solutions quickly is an essential skill, as it saves time in graphing analysis and problem-solving.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so that its slope and intercept are immediately clear. The general formula is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

The slope \(m\) indicates the steepness of the line and the direction it slants. If \(m\) is positive, the line ascends from left to right; if negative, it descends. The y-intercept \(b\) is where the line crosses the y-axis.

In practice, converting equations to slope-intercept form helps simplify graphing and identifying relationships between lines. For the given exercise, both equations resulted in the form \(y = \frac{3}{2}x - \frac{3}{2}\), showing they are the same line with the same slope and intercept.

Key advantages of the slope-intercept form include:

• Easy visualization of lines for graphing.
• Simplified calculation of intersections and parallels.
• Immediate recognition of infinite solutions when comparing line equations.

Mastering the transition to slope-intercept form can greatly aid in understanding complex systems of equations.