Problem 25 - Intermediate Algebra

Question

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. See Examples 3 and 4 $$ \left\\{\begin{array}{l} 3 x-3 y=4 \\ x-y=4 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent; the lines are parallel.

Step 1: Identify the equations

The system of equations given is:\( 3x - 3y = 4 \) and \( x - y = 4 \).

Step 2: Convert equations to slope-intercept form

Convert each equation to the slope-intercept form \( y = mx + b \). For the first equation, divide all terms by 3 to get \( x - y = \frac{4}{3} \). Adding \( y \) to both sides, rearrange it to \( y = x - \frac{4}{3} \). For the second equation, rearrange to \( y = x - 4 \).

Step 3: Graph the first equation

The equation \( y = x - \frac{4}{3} \) has a slope of 1 and a y-intercept of \(-\frac{4}{3}\). Plot the y-intercept on the graph and use the slope to find another point, and then draw the line.

Step 4: Graph the second equation

The equation \( y = x - 4 \) has a slope of 1 and a y-intercept of \(-4\). Plot the y-intercept and use the slope to find another point, then draw this line on the graph.

Step 5: Analyze the graph

The lines represented by \( y = x - \frac{4}{3} \) and \( y = x - 4 \) are parallel because they have the same slope but different y-intercepts. Parallel lines do not intersect, meaning the system is inconsistent.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves representing an equation graphically on a coordinate plane. This process makes finding solutions to the equations simpler. Each equation is represented by a line, with the relationship between the variables dictated by the line's slope and y-intercept. If a system of equations comprises two or more linear equations, the solution is the point where the lines intersect.
In order to graph a linear equation, you need:

• The slope (9392) of the line, which represents the steepness and direction.
• The y-intercept, which is where the line crosses the y-axis.

By plotting the y-intercept, and using the slope to determine another point on the line, you can easily draw the line representing the equation. This method allows you to visualize the relationships in a system of equations, and assess whether they have a common solution.

Slope-Intercept Form

The slope-intercept form of a linear equation is a convenient way to express the equation of a line. It takes the form:
\( y = mx + b \)
This format allows you to immediately identify two important characteristics of the line:

• Slope (97): The coefficient of \(x\), which signifies how much \(y\) changes for a unit increase in \(x\).
• Y-intercept (b): The value of \(y\) when \(x = 0\).

Both values play a critical role in graphing. With the y-intercept, you have a starting point on the graph, and with the slope, you can plot subsequent points by moving along the line. By converting the given equations into the slope-intercept form, you can easily graph them and determine their points of intersection.

Inconsistent Systems

An inconsistent system refers to a set of equations with no common solution. This typically happens when the equations represent parallel lines. Even though parallel lines have the same slope, their different y-intercepts prevent them from ever meeting. Therefore, an inconsistent system essentially describes two equations that travel along the same path without overlap.
This can often be identified quickly by converting each equation in a system to the slope-intercept form. If the resulting slopes are identical but the y-intercepts differ, the system lacks a solution. Visualizing this through graphing further clarifies the impossibility of intersection, thus confirming the inconsistency.

Parallel Lines in Algebra

In algebra, parallel lines are lines located on the same plane that will never meet, regardless of how far they extend. You can easily identify parallel lines when graphing equations by comparing their slopes. If two lines have the same slope but different y-intercepts, they are parallel.

• This implies they will never cross each other and thus have no points in common.
• You can spot this characteristic when an equation is in slope-intercept form (64 = 81x + b).

Recognizing parallel lines is essential when solving systems of equations, as it indicates an inconsistent system. By understanding this concept, students can better interpret the graphical representation of solutions.