Problem 21 - Algebra 2

Question

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(x+y=4\) \(-4 x+y=9\)

Step-by-Step Solution

Answer

The system is consistent and independent.

Step 1: Identify the Equations

The system of equations given for this problem is: 1. \( x + y = 4 \) 2. \( -4x + y = 9 \).

Step 2: Rewrite Equations in Slope-Intercept Form

The slope-intercept form of an equation is \( y = mx + b \). Let's rewrite both equations in this form. - **Equation 1:** \( x + y = 4 \) Solving for \( y \), we get \( y = -x + 4 \). - **Equation 2:** \( -4x + y = 9 \) Solving for \( y \), we get \( y = 4x + 9 \).

Step 3: Graph the Equations

Graph each equation on the same coordinate plane. - **Equation 1:** \( y = -x + 4 \) This line has a slope of \(-1\) and a y-intercept of \(4\). - **Equation 2:** \( y = 4x + 9 \) This line has a slope of \(4\) and a y-intercept of \(9\). Draw both lines on a graph to see where they intersect.

Step 4: Analyze the Intersection

Observe the graph for any point of intersection between the two lines. The intersection point represents the solution to the system, if it exists.

Step 5: Determine System Consistency and Dependency

Since the two lines intersect at a point, the system is consistent and independent. Being consistent means there is at least one solution, and independent means the solution is unique. The point of intersection can be determined by solving the equations algebraically or graphically.

Key Concepts

Graphing Linear Equations

To solve a system of equations graphically, you need to draw both equations on the same coordinate plane. This method gives a visual representation of where the two equations meet. Essentially, each equation forms a line, and the point where these lines intersect is the solution.

When you graph linear equations:

• Determine the slope and y-intercept using the slope-intercept form of the equation.
• Plot the y-intercept on the graph.
• Use the slope to find another point on the line. Remember that slope is the change in y over the change in x.

Draw the line through these points with a ruler for accuracy. Repeat these steps for the second equation. This forms two lines on the graph, and your goal is to find their intersection point.

Consistent and Independent Systems

In systems of equations, understanding terms like consistent and independent helps in determining the nature of the solutions. A system is consistent if it has at least one solution, and it is independent if it exactly has one unique solution, which means the lines intersect at only one point on the graph.

Here's how to identify them:

• A consistent system means the lines do intersect.
• An independent system means the intersection is a single point.
• An inconsistent system means the lines do not intersect (they are parallel).
• A dependent system means the lines coincide, resulting in infinite intersections and solutions.

In the given example, the system is consistent and independent since the lines intersect at exactly one point, providing one unique solution.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of expressing the line's equation so it's easier to graph. The form is given by the equation: \[ y = mx + b \]where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, which is where the line crosses the y-axis.

By converting equations into this form, you can quickly identify how steep a line is and where it starts on the graph. For example: - Equation 1: \( y = -x + 4 \) with a slope of -1 and a y-intercept of 4.- Equation 2: \( y = 4x + 9 \) with a slope of 4 and a y-intercept of 9. These values make plotting the lines straightforward, helping you see where they might intersect.

Solutions to Systems of Equations

The solution to a system of linear equations is the point where the lines intersect each other on the graph. This represents a set of values that satisfy all equations in the system.

Once the lines are drawn using the slope-intercept form, look for the meeting point on the graph. This intersection indicates a shared solution:

• If lines intersect at a single point, there's one unique solution.
• If they don't intersect (are parallel), there's no solution.
• If the lines overlap completely, there are infinite solutions.

In the example of equations given, solving logically or graphically shows where the solutions lie. This helps in understanding whether the system is consistent (having solutions) and independent (one precise solution).