Problem 54 - Algebra 2

Question

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(3 x+y=1\) \(y=2 x-4\)

Step-by-Step Solution

Answer

The system is consistent and independent because the lines intersect at one point.

Step 1: Convert equations to slope-intercept form

The first equation is already in standard form: \(3x+y=1\). To convert it to slope-intercept form (\(y=mx+b\)), solve for \(y\). Thus, \(y = -3x + 1\). The second equation is already in slope-intercept form: \(y = 2x - 4\). Now both equations are in slope-intercept form.

Step 2: Identify slopes and y-intercepts

From the first equation \(y = -3x + 1\), the slope \(m_1\) is \(-3\) and the y-intercept is \(1\). For the second equation \(y = 2x - 4\), the slope \(m_2\) is \(2\) and the y-intercept is \(-4\).

Step 3: Compare slopes to determine relationship

Since the slopes of the two lines \(-3\) and \(2\) are different, the lines intersect at one point, meaning the system is consistent and independent.

Step 4: Graph the equations

Plot the line \(y = -3x + 1\) using the y-intercept \(1\) and the slope \(-3\) (which goes down 3 units for every 1 unit to the right). Then, plot the line \(y = 2x - 4\) using the y-intercept \(-4\) and the slope \(2\) (which goes up 2 units for every 1 unit to the right).

Step 5: Identify intersection point

After graphing the lines, identify the intersection point visually on the graph. This point satisfies both equations and represents the solution to the system.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is an extremely useful way to express a line algebraically. It is generally written as \( y = mx + b \), where:

• \( m \) is the slope of the line, indicating how steep the line is.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

To convert an equation from standard form (like \( Ax + By = C \)) to slope-intercept form, you solve for \( y \). For instance, by rearranging \( 3x + y = 1 \), we get \( y = -3x + 1 \). Now, both the slope and y-intercept are clear: the slope \( m = -3 \) and the y-intercept \( b = 1 \). This step is crucial for graphing equations easily and helps in visualizing the behavior of the line.

Consistent and Independent Systems

A system of equations can be classified by analyzing the relationship between lines represented by the equations. To classify the system:

• Consistent: This means the system has at least one solution. The lines intersect at one or more points.
• Independent: This indicates that the system has exactly one solution, meaning the lines intersect at exactly one point.

When you have a consistent and independent system, like in the equations \( y = -3x + 1 \) and \( y = 2x - 4 \), the lines will intersect at exactly one point. Therefore, this scenario shows that the lines are neither parallel nor identical, which results in one intersection point and a single solution. This classification is essential for solving and interpreting solutions effectively, particularly when graphing linear equations.

Graphing Linear Equations

Graphing linear equations involves plotting points on a two-dimensional plane to represent the equations visually. This method helps to see the relationship between two variables clearly. Here's how to graph an equation like \( y = mx + b \):

• Start by plotting the y-intercept \( b \) on the y-axis.
• Use the slope \( m \) to find the next points. The slope \( m \) tells you how to move from the y-intercept. For example, a slope of \(-3\) means moving down 3 units for every unit you move right.
• Draw the line that connects these points.

When graphing two equations, as done with \( y = -3x + 1 \) and \( y = 2x - 4 \), you can visually identify their intersection point. This point is the solution to the system of equations. By graphing, you simplify the understanding of linear relationships and find solutions through visual means, which is especially helpful for those who are visual learners.