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Chapter 9: Problem 36

Solve each system. State whether it is inconsistent or has infinitely many solutions. If the system has infinitely many solutions, write the solution set with y arbitrary. $$\begin{array}{r} 2 x-3 y-7=0 \\ -4 x+6 y-14=0 \end{array}$$

Short Answer

Expert verified
The system has infinitely many solutions. The solution set is \((x, y) = \left( \frac{3y + 7}{2}, y \right)\).

Step by step solution

01

Write the system of equations

Given the system of equations: 1) \(2x - 3y - 7 = 0\) 2) \(-4x + 6y - 14 = 0\)
02

Simplify the second equation

Notice that the second equation is \(-4x + 6y - 14 = 0\). This can be simplified by dividing the entire equation by -2:\(\frac{-4x}{-2} + \frac{6y}{-2} - \frac{14}{-2} = 0\) which simplifies to: \(2x - 3y - 7 = 0\)
03

Compare the equations

The simplified second equation \(2x - 3y - 7 = 0\) is exactly the same as the first equation. Therefore, the two equations are actually representing the same line.
04

Determine the solution

Since the two equations are identical, the system does not have just one solution or no solution. It has infinitely many solutions. Any point on the line \(2x - 3y - 7 = 0\) is a solution.
05

Express the solution

To write the solution set with \(y\) arbitrary, solve the first equation for \(x\): \(2x - 3y - 7 = 0\) \(2x = 3y + 7\) \(x = \frac{3y + 7}{2}\) Therefore, the solution set is: \((x, y) = \left( \frac{3y + 7}{2}, y \right)\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Consistency of Linear Systems
When solving systems of linear equations, one of the first steps is determining the consistency of the system. A system is consistent if there is at least one solution that satisfies all equations in the system.
An inconsistent system, on the other hand, has no solution at all.
To test for consistency, we check if the equations represent the same plane or intersect at a common point.
In our exercise, both equations simplify to the same line, meaning they are not contradictory. Thus, the system in the exercise is consistent because it has solutions.
Infinitely Many Solutions
A system with infinitely many solutions means that every point on a particular line satisfies both equations.
This usually happens when the two equations are essentially the same.
Our exercise reveals that the same line appears twice, indicating that the system does not have a unique intersection point.
Instead, every point on this line is a solution to our system of equations.
This outcome occurs when the equations are multiples of each other, ensuring they are identical.
Arbitrary Variable in Solutions
In systems with infinitely many solutions, we often describe the solution set by using an arbitrary variable.
In our solution, we solved one of the equations for x in terms of y:
\[ x = \frac{3y + 7}{2} \]
Here, y is an arbitrary variable that can take any value.
This form allows us to express any solution to the system. By choosing different values for y, we get corresponding values of x, all of which lie on the line represented by the equations.
Linear Equation Simplification
Simplification helps in identifying relationships between equations.
In our exercise, the second equation simplifies by dividing it by -2:
\[-4x + 6y - 14 = 0\]
becomes
\[ 2x - 3y - 7 = 0 \]
which is identical to the first equation.
This simplification shows that both equations are the same, proving they represent the same line.
Simplifying equations can reveal hidden consistencies or inconsistencies, helping us accurately determine the solution set.

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Most popular questions from this chapter

Find the equation of the circle passing through the given points. $$(-1,3),(6,2), \text { and }(-2,-4)$$

Solve each problem. Plate-Glass Sales The amount of plate-glass sales \(S\) (in millions of dollars) can be affected by the number of new building contracts \(B\) issued (in millions) and automobiles \(A\) produced (in millions). A plate-glass company in California wants to forecast future sales by using the past three years of sales. The totals for the three years are given in the table. To describe the relationship among these variables, we can use the equation $$ S=a+b A+c B $$. where the coefficients \(a, b,\) and \(c\) are constants that must be determined before the equation can be used. (Source: Makridakis, S., and S. Wheelwright, Forecasting Methods for Management, John Wiley and Sons.) (a) Substitute the values for \(S, A,\) and \(B\) for each year from the table into the equation \(S=a+b A+c B,\) and obtain three linear equations involving \(a, b,\) and \(c\) (b) Use a graphing calculator to solve this linear system for \(a, b,\) and \(c .\) Use matrix inverse methods. (c) Write the equation for \(S\) using these values for the coefficients. $$\begin{array}{|c|c|c|} \hline S & A & B \\ \hline 602.7 & 5.543 & 37.14 \\\ \hline 656.7 & 6.933 & 41.30 \\ \hline 778.5 & 7.638 & 45.62 \\ \hline \end{array}$$ (d) For the next year it is estimated that \(A=7.752\) and \(B=47.38 .\) Predict \(S .\) (The actual value for \(S\) was \(877.6 .\) ) (e) It is predicted that in 6 yr, \(A=8.9\) and \(B=66.25 .\) Find the value of \(S\) in this situation and discuss its validity.

For each pair of matrices \(A\) and \(B,\) find \((a) A B\) and \((b) B A\). $$A=\left[\begin{array}{rr} 0 & -5 \\ -4 & 2 \end{array}\right], B=\left[\begin{array}{rr} 3 & -1 \\ -5 & 4 \end{array}\right]$$

Use the determinant theorems to find the value of each determinant. $$\left|\begin{array}{rrr} 9 & 1 & 7 \\ 12 & 5 & 2 \\ 11 & 4 & 3 \end{array}\right|$$

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