Problem 20 - Algebra 2

Question

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(y=x+3\) \(y=2 x+6\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Equations

We have two equations: \(y = x + 3\) and \(y = 2x + 6\). These equations are linear, representing straight lines.

Step 2: Determine the Slope and Y-Intercept

For the first equation \(y = x + 3\), the slope \(m\) is 1, and the y-intercept \(b\) is 3.For the second equation \(y = 2x + 6\), the slope \(m\) is 2, and the y-intercept \(b\) is 6.

Step 3: Graph the Equations Using Slope and Y-Intercepts

Start by plotting the y-intercept for both lines. For \(y = x + 3\), plot the point (0, 3). For \(y = 2x + 6\), plot the point (0, 6). Then use the slopes to find another point for each line. For \(y = x + 3\), move up 1 and right 1 for another point. For \(y = 2x + 6\), move up 2 and right 1 for another point. Draw the lines.

Step 4: Analyze the Graph

The graph shows two distinct lines intersecting. If two lines intersect at exactly one point, they are consistent and independent.

Key Concepts

Graphing Equations

Graphing equations is like drawing a picture of a math problem. For each equation, especially linear ones, you put points on a graph which show where the line should go. Here's how to tackle it:

• First, find the y-intercept which is the point where the line crosses the y-axis.
• Then, use the slope to move from this point, plotting additional points along the line.
• Connect the dots to draw the line.

By graphing each line on the same coordinate plane, you can visually see how they relate to each other. This visualization helps determine the relationship between the equations.

Linear Equations

Linear equations are equations where the highest power of the variable is one. These equations graph as straight lines. Both equations from our exercise, \(y = x + 3\) and \(y = 2x + 6\), are linear. Key features of linear equations include:

• They only involve addition, subtraction, and multiplication of variables by constants.
• Their graph is a straight line, no curves or fancy shapes!
• Understanding the slope and y-intercept helps in graphing them easily.

Linear equations are quick to work with because they have predictable changes, shown by their constant slope.

Consistent and Independent Systems

When examining systems of equations, understanding their solution type is crucial. A system is labeled as consistent and independent if its equations intersect at exactly one point. Here's what this means:

• "Consistent" means there is at least one solution.
• "Independent" indicates the solution is unique (one intersection point).
• In our exercise, the two lines intersect at a single point, so the system is both consistent and independent.

This classification helps in understanding the nature of the solutions to these systems of equations.

Slope and Y-Intercept

Slope and y-intercept are essential features for understanding and graphing linear equations.

Slope

The slope tells you how steep a line is and its direction. For the equation \(y = mx + b\), "m" represents the slope.

• Positive slopes rise from left to right.
• Negative slopes fall from left to right.

For \(y = x + 3\), the slope is 1. For \(y = 2x + 6\), the slope is 2.

Y-Intercept

The y-intercept is where the line crosses the y-axis, represented by "b" in the equation \(y = mx + b\).

• For \(y = x + 3\), it crosses at 3.
• For \(y = 2x + 6\), it crosses at 6.

Together, slope and y-intercept make graphing straightforward and quick.

Analyzing Graph Intersection

Analyzing the point where two graphs intersect is key in determining relationships in systems of equations.

• Intersection points tell you where the solutions of the system exist.
• If lines intersect, as in our example, the point signifies a unique solution (consistent and independent).
• Multiple intersection points or none indicate different system classifications (e.g., consistent and dependent or inconsistent).

By clearly identifying and examining these intersections, you can understand much more than just the equations themselves—it's about their interaction.