Problem 9 - Algebra 2

Question

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(x+2 y=2\) \(2 x+4 y=8\)

Step-by-Step Solution

Answer

The system is inconsistent because the lines are parallel with different y-intercepts.

Step 1: Re-write in Slope-Intercept Form

To begin with, convert each equation to the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.For the first equation \(x + 2y = 2\):\[ 2y = -x + 2 \] \[ y = -\frac{1}{2}x + 1 \]For the second equation \(2x + 4y = 8\):\[ 4y = -2x + 8 \]\[ y = -\frac{1}{2}x + 2 \]

Step 2: Analyze the Slopes and Intercepts

Now that both equations are in slope-intercept form, compare their slopes and intercepts.- Both lines have a slope of \(-\frac{1}{2}\), which indicates that they are parallel.- The y-intercepts are different: 1 and 2 for the first and second equations, respectively. Hence, they never intersect.

Step 3: Consistency and Dependency Check

Since the two lines are parallel and have different y-intercepts, there are no points of intersection. - Therefore, the system of equations is inconsistent because there is no solution to the system.

Key Concepts

Slope-Intercept Form

Observing mathematical equations in the slope-intercept form reveals the equation's slope and y-intercept, which are vital for plotting graphs. A linear equation written as \(y = mx + b\), where \(m\) represents the slope and \(b\) symbolizes the y-intercept, is recognized as being in slope-intercept form.
**How to Convert:**

• Start with the linear equation, often written as \(Ax + By = C\).
• Rearrange it to solve for \(y\).
• Make \(y\) the subject of the equation: divide by \(B\) post rearrangement.
This format offers a clear understanding of a line's steepness and the point where it crosses the y-axis.

Parallel Lines

Parallel lines are two lines in a plane that never meet. They are always the same distance apart and will never intersect. When analyzing systems of equations, identifying parallel lines is essential.
**Characteristics of Parallel Lines:**

• They have the same slope, indicated as \(m\) in their equations.
• The y-intercepts differ, meaning they are not just a single line.
• Since they do not intersect, they do not have a solution in common.

These evident properties help determine when a system of equations has no solution, which makes it an inconsistent system.

Systems of Equations

A system of equations consists of two or more equations that share the same set of variables. We solve systems to find a common solution or to see if there is a discrepancy. There are key types of systems that define the relationships of their equations.
**Types of Systems:**

• **Consistent Systems:** There is at least one set of values that satisfies all equations.
• **Independent Systems:** Possess exactly one solution due to intersecting lines.
• **Dependent Systems:** Lines overlap completely, representing the same equation, having infinite solutions.
• **Inconsistent Systems:** No set of values can solve all equations simultaneously; such lines run parallel.

Solution of a System

Finding the solution of a system involves determining the point(s) where all equations are satisfied simultaneously. The solution is the intersection point of the graphs of the equations.
**Possible Outcomes for Solutions:**

• **Single Solution:** When lines intersect at one point, giving exactly one solution.
• **Infinite Solutions:** Occurs when lines overlap completely, showing they are the same mathematically.
• **No Solution:** Happens with parallel lines that do not meet at any point, leading to inconsistency.

Determining these points is crucial, as it provides meaningful insights into relationships contained in the equations.