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Chapter 6: Problem 61

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}&3 x+2 y=4\\\&6 x+4 y=8\end{aligned}$$

Short Answer

Expert verified
The system is dependent, with solutions \((x, y) = (t, 2 - \frac{3}{2}t)\), where \(t\) is any real number.

Step by step solution

01

Set Up Coefficient Matrix

First, identify the coefficient matrix from the system of equations. For the equations \(3x + 2y = 4\) and \(6x + 4y = 8\), the coefficient matrix is \(A = \begin{bmatrix} 3 & 2 \ 6 & 4 \end{bmatrix}\).
02

Compute Determinant of A

Calculate the determinant of the coefficient matrix \(A\). The determinant \(D\) is given by \(D = \begin{vmatrix} 3 & 2 \ 6 & 4 \end{vmatrix} = (3)(4) - (2)(6) = 12 - 12 = 0\).
03

Analyze Determinant

Since the determinant \(D = 0\), Cramer's Rule cannot be used to solve the system directly because this indicates that the equations are either dependent (same line) or inconsistent (parallel lines).
04

Examine Equation Consistency

To determine whether the system is dependent or inconsistent, rewrite the second equation \(6x + 4y = 8\) in terms of the first: divide by 2 to obtain \(3x + 2y = 4\), which is identical to the first equation, implying the system is dependent.
05

State Solution in Parametric Form

Since the system is dependent, it has infinitely many solutions. Express \(x\) or \(y\) in terms of a parameter, say \(t\). Solving \(3x + 2y = 4\) for \(y\) gives \(y = 2 - \frac{3}{2}x\). Let \(x = t\), then \(y = 2 - \frac{3}{2}t\), giving the solution \((x, y) = (t, 2 - \frac{3}{2}t)\).
06

Verify and Interpret Solution

Verify by substituting \((x, y) = (t, 2 - \frac{3}{2}t)\) into either equation, confirming both equations hold true for any real number \(t\). The system has infinitely many solutions given parametrically by \((x, y) = (t, 2 - \frac{3}{2}t)\).

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Key Concepts

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

Determinant of a Matrix
When solving systems of linear equations with Cramer's Rule, the determinant of the coefficient matrix is crucial. The determinant is a special number calculated from a square matrix. For a 2x2 matrix, say \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant \(D\) is computed as \(D = ad - bc\). This value helps decide whether the system has a unique solution or not. If \(D eq 0\), Cramer's Rule can be applied directly to solve for the variables. However, if \(D = 0\), as in our given problem, Cramer's Rule is not applicable and you have a special case to consider.
Dependent Equations
Dependent equations refer to equations in a system that are multiples of each other. Essentially, they represent the same line in a coordinate plane. This characteristic means they have the same slope and intercept, leading to them lying exactly on top of one another. In the exercise, the equations \(3x + 2y = 4\) and \(6x + 4y = 8\) are dependent. This is because the second equation can be simplified, by dividing every term by 2, to become identical to the first one. Whenever equations are dependent, their graphs overlap completely, indicating a unique set of points that satisfy both equations.
Parametric Form
In cases where equations in a system are dependent, and thus have more than one solution, the solutions can often be written in parametric form. This involves expressing one variable in terms of another variable, often called the parameter (here denoted as \(t\)). In our exercise, for the equation \(3x + 2y = 4\), if we solve for \(y\) in terms of \(x\), we get \(y = 2 - \frac{3}{2}x\). By setting \(x = t\), we can express \(y\) as \(y = 2 - \frac{3}{2}t\). Hence, the solutions to the system of equations can be described parametrically as \((x, y) = (t, 2 - \frac{3}{2}t)\).
Infinitely Many Solutions
When you have dependent equations, as described above, the system doesn't have a unique solution but rather infinitely many solutions. This means any value of the parameter will give you a valid solution that satisfies the original system of equations. For the system given in the exercise, no matter what real number \(t\) is chosen for \(x\), you can calculate a corresponding \(y\), resulting in an infinite set of solutions \((t, 2 - \frac{3}{2}t)\). This outcome is a clear indication that the original two equations describe the same line, and thus share all their points. Such a system exhibits an infinite number of solutions, as all points on the line are solutions.

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Most popular questions from this chapter

Solve each nonlinear system of equations analytically. $$\begin{aligned}x^{2}+y^{2} &=10 \\\\-x^{2}+y &=-4\end{aligned}$$

As the price of a product increases, businesses usually increase the quantity manufactured. However, as the price increases, consumer demand-or the quantity of the product purchased by consumers-usually decreases. The price we see in the market place occurs when the quantity supplied and the quantity demanded are equal. This price is called the equilibrium price and this demand is called the equilibrium demand. The supply of a certain product is related to its price by the equation \(p=\frac{1}{3} q,\) where \(p\) is in dollars and \(q\) is the quantity supplied in hundreds of units. (a) If this product sells for 9 dollars, what quantity will be supplied by the manufacturer? (b) Suppose that consumer demand for the same product decreases as price increases according to the equation \(p=20-\frac{1}{5} q .\) If this product sells for 9 dollars, what quantity will consumers purchase? How does this compare with the quantity being supplied by the manufacturer at this price? (c) On the basis of parts (a) and (b), what should happen to the price? Explain. (d) Determine the equilibrium price at which the quantity supplied and quantity demanded are equal. What is the demand at this price?

Find a system of linear inequalities for which the graph is the region in the first quadrant between and inclusive of the pair of lines \(x+2 y-8=0\) and \(x+2 y=12\)

The table shows weight \(W,\) neck size \(N,\) overall length \(L,\) and chest size \(C\) for four bears. $$\begin{array}{|c|c|c|c|} \hline W \text { (pounds) } & N \text { (inches) } & L \text { (inches) } & C \text { (inches) } \\ \hline 125 & 19 & 57.5 & 32 \\\ 316 & 26 & 65 & 42 \\ 436 & 30 & 72 & 48 \\ 514 & 30.5 & 75 & 54 \end{array}$$ A. We can model these data with the equation $$ W=a+b N+c L+d C $$ where \(a, b, c,\) and \(d\) are constants. To do so, represent a system of linear equations by a \(4 \times 5\) augmented matrix whose solution gives values for \(a, b, c,\) and \(d\) B. Solve the system. Round each value to the nearest thousandth. C. Predict the weight of a bear with \(N=24, L=63\) and \(C=39 .\) Interpret the result.

Solve each application. A company produces two models of bicycles: model \(A\) and model \(B\). Model \(A\) requires 2 hours of assembly time, and model \(B\) requires 3 hours of assembly time. The parts for model \(A\) cost \(\$ 25\) per bike; those for model \(B\) cost \(\$ 30\) per bike. If the company has a total of 34 hours of assembly time and \(\$ 365\) available per day for these two models, what is the maximum number of each model that can be made in a day and use all of the available resources?
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