Problem 55 - Intermediate Algebra

Question

Solve each system by graphing, if possible. If a system is inconsistent or if the equations are dependent, state this. (Hint: Several coordinates of points of intersection are fractions.) $$ \left\\{\begin{array}{l} x=13-4 y \\ 3 x=4+2 y \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is the point \((23, -\frac{5}{2})\); the system is consistent and independent.

Step 1: Graph the first equation

The first equation is given as \(x = 13 - 4y\). We can rewrite this in terms of \(y\) to make graphing easier: \(y = \frac{13 - x}{4}\). To graph, choose values for \(y\) and calculate corresponding \(x\). For example, if \(y = 0\), then \(x = 13\). If \(y = 1\), then \(x = 9\). Plot these points and draw the line.

Step 2: Graph the second equation

The second equation is \(3x = 4 + 2y\). Again, express \(y\) in terms of \(x\) for easier graphing: \(y = \frac{3x - 4}{2}\). Choose values for \(x\) and calculate \(y\). For example, if \(x = 2\), \(y = 2\). If \(x = 0\), \(y = -2\). Plot these points and draw this line on the graph.

Step 3: Find the point of intersection

To find the intersection point graphically, look for where the two lines cross on the graph. According to the equations, substitute a value of \(y = -\frac{5}{2}\) (found by solving algebraically) into \(x = 13 - 4y\) to find \(x = 23\), and check by substitution into the second equation \(3x = 4 + 2y\) to confirm intersection point is \((23, -\frac{5}{2})\). The lines intersect at exactly this point.

Step 4: Check for consistency or dependency

Since the lines intersect at exactly one point, the system is consistent and independent. There is no indication of parallelism or the same line, confirming different slopes and intersection.

Key Concepts

Inconsistent Systems

In the context of solving systems of linear equations, an inconsistent system is one where there are no solutions. This happens when the equations represent parallel lines on the graph. Since parallel lines never intersect, there is no common solution that satisfies both equations.

Recognizing an inconsistent system involves examining the slopes of the lines in your equations. If two lines have the same slope but different y-intercepts, they are parallel. To see this clearly:

• Convert the given equations into the slope-intercept form, i.e., \(y = mx + b\), where \(m\) is the slope.
• Compare the slopes of both lines.
• If \(m_1 = m_2\) and the constants \(b_1 eq b_2\), then you're dealing with an inconsistent system.

In our original exercise, the two lines represented by the equations intersect at one point, meaning the system is consistent.

Dependent Equations

Dependent equations are those equations in a system that represent the same line on a graph. This means that both equations result in an identical line, making every point on the line a solution to the system.

To identify dependent equations, look for equations that reduce to the same relationship. You can use these methods:

• Simplify both equations; if they become equivalent, they are dependent.
• Check if one equation is a multiple of the other.

In a graphical solution, overlapping lines indicate dependent equations. This would mean an infinite number of solutions, as every point on the line satisfies both equations. In the given exercise, however, the equations resulted in distinct lines, thus they were independent.

Graphical Representation

Graphical representation is a powerful visualization tool used to solve systems of equations. By plotting equations on a graph, you can visually identify the solution as points of intersection.

Here's how you can graphically represent a system of linear equations:

• Convert each equation to the slope-intercept form (\(y = mx + b\)) for easier plotting.
• Choose different values for \(x\) or \(y\) to calculate corresponding coordinates.
• Plot these coordinates on the graph.
• Draw lines through the plotted points for each equation.

This method not only helps to find the solution efficiently but also aids in understanding the nature of the system (consistent, inconsistent, independent, or dependent). In the exercise, we saw that the graphical intersection of the lines resulted in one single point, highlighting a unique solution.

Substitution Method

The substitution method is a straightforward algebraic technique to solve systems of linear equations. This involves solving one equation for one variable and substituting it into another equation.

Here's a simple guide to using the substitution method:

• Pick one of the equations and solve it for one variable (e.g., solve \(x = 13 - 4y\) for \(x\)).
• Substitute this expression into the other equation (replace \(x\) in \(3x = 4 + 2y\) to find \(y\)).
• Solve the new equation for the remaining variable.
• Substitute back to find the other variable's value.

The substitution method can provide precise values and is particularly helpful when graphical methods result in complex or fractional intercepts. In the original example, substitution was used to verify the intersection point \(23, -\frac{5}{2}\), ensuring accuracy.