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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 40 (page 729)

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.
$3 x + 3 y = 3 4 x + 2 y = \frac{8}{3}$

Short Answer

Expert verified

The required solution is $(x, y) = (\frac{1}{3}, \frac{2}{3})$

Step by step solution

01

Step 1. Given Information

The given equations are

$3 x + 3 y = 3 4 x + 2 y = \frac{8}{3}$

02

Step 2. Calculation

We will first write the augmented matrix, by writing constants on right side of vertical bar.

$[\begin{matrix} 3 & 3 | & 3 \\ 4 & 2 | & \frac{8}{3} \end{matrix}]$

Now, we will make the entry 0 in row 2 column 1 by performing the row operations $R_{1} = \frac{1}{3} R_{1}$ and then apply $R_{2} = - 4 R_{1} + R_{2}$

localid="1647367575351" $[\begin{matrix} 3 & 3 | & 3 \\ 4 & 2 | & \frac{8}{3} \end{matrix}] = [\begin{matrix} \frac{3}{3} & \frac{3}{3} | & \frac{3}{3} \\ 4 & 2 | & \frac{8}{3} \end{matrix}] = [\begin{matrix} 1 & 1 | & 1 \\ 4 & 2 | & \frac{8}{3} \end{matrix}] = [\begin{matrix} 1 & 1 | & 1 \\ - 4 (1) + 4 & - 4 (1) + 2 | & - 4 (1) + \frac{8}{3} \end{matrix}] = [\begin{matrix} 1 & 1 | & 1 \\ 0 & - 2 | & \frac{- 4}{3} \end{matrix}]$

03

Step 3. Calculation

Use the obtained matrix to write the system of equations.

$x + y = 1 - 2 y = - \frac{4}{3}$

Solve the equations to find the solution set.

$y = \frac{- 4}{3} 脳 \frac{- 1}{2} 鈬� y = \frac{2}{3} x + \frac{2}{3} = 1 x = \frac{1}{3}$

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Most popular questions from this chapter

Verify that the values of the variables listed are solutions of the system of equations.

$\{\begin{matrix} \begin{array}{rcl} 3 x + 3 y + 2 z & = & 4 \\ x - y - z & = & 0 \end{array} \\ 2 y - 3 z = - 8 \end{matrix} x = 1, y = - 1, z = 2; (1, - 1, 2)$

Solve each system of equations. If the system has no solution, say that it is inconsistent

x + 2y - z = -3

2x - 4y + z = -7

-2x + 2y - 3z = 4

In Problems 23鈥�34, graph each system of linear inequalities by hand. Verify your results using a graphing utility:

$\{\begin{cases} 2 x + 3 y 鈮� 6 \\ 2 x + 3 y 鈮� 0 \end{cases}$

If the solution to a system of two linear equations containing two unknowns is $x = 3, y = - 2$ , then the lines intersect at the point _________.

Graph each inequality by hand.

$2 x + y 鈮� 6$

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