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Chapter 4: Q39. (page 158)

Solve each system of equations.
$\begin{matrix} 3 r - 15 s + 4 t = - 57 \\ 9 r + 45 s - t = 26 \\ - 6 r + 10 s + 3 t = - 19 \end{matrix}$

Short Answer

Expert verified

The solution of given system of equation is $(- \frac{4}{3}, \frac{3}{5}, - 11)$ .

Step by step solution

01

– Description of step.

Add $9 r + 45 s - t = 26$ and $- 6 r + 10 s + 3 t = - 19$ .

\begin{matrix} 9 r + 45 s 鈭� t 鈭� 6 r + 10 s + 3 t = 26 鈭� 19 \\ 3 r + 55 s + 2 t = 7 \end{matrix}

02

– Description of step.

Subtract $3 r - 15 s + 4 t = - 57$ from the obtained equation in step 1 then divide each side of the obtained equation by 2 and solve for $s$ .

\begin{matrix} 3 r + 55 s + 2 t 鈭� (3 r 鈭� 15 s + 4 t) = 7 鈭� (鈭� 57) \\ 3 r + 55 s + 2 t 鈭� 3 r + 15 s 鈭� 4 t = 64 \\ \frac{70}{2} s 鈭� \frac{2}{2} t = \frac{64}{2} \\ 35 s 鈭� t = 32 \\ s = \frac{t}{35} + \frac{32}{35} \end{matrix}

03

– Description of step.

Multiply $3 r - 15 s + 4 t = - 57$ by 3 and add obtained equation to $9 r + 45 s - t = 26$ then solve for $r$ .

\begin{matrix} 9 r 鈭� 45 s + 12 t + 9 r + 45 s 鈭� t = 鈭� 171 + 26 \\ 18 r + 11 t = 鈭� 145 \\ 18 r = 鈭� 145 鈭� 11 t \\ r = 鈭� \frac{145}{18} 鈭� \frac{11}{18} t \end{matrix}

04

– Description of step.

Substitute $\frac{t}{35} + \frac{32}{35}$ for $s$ and $- \frac{145}{18} - \frac{11}{18} t$ for $r$ into $- 6 r + 10 s + 3 t = - 19$ find the value of $t$ .

\begin{matrix} 鈭� 6 (鈭� \frac{145}{18} 鈭� \frac{11}{18} t) + 10 (\frac{t}{35} + \frac{32}{35}) + 3 t = 鈭� 19 \\ (\frac{11}{3} + \frac{2}{7} + 3) t = 鈭� 19 鈭� \frac{145}{3} 鈭� \frac{64}{7} \\ \frac{146}{21} t = 鈭� \frac{1606}{21} \\ t = 鈭� \frac{1606}{146} \\ = 鈭� 11 \end{matrix}

05

– Description of step.

Substitute $- 11$ for $t$ into $s = \frac{t}{35} + \frac{32}{35}$ and simplify.

\begin{matrix} s = \frac{鈭� 11}{35} + \frac{32}{35} \\ = \frac{鈭� 11 + 32}{35} \\ = \frac{3}{5} \end{matrix}

06

– Description of step.

Substitute $- 11$ for $t$ into $r = - \frac{145}{18} - \frac{11}{18} t$ and simplify.

\begin{matrix} r = 鈭� \frac{145}{18} 鈭� \frac{11}{18} 脳 (鈭� 11) \\ = \frac{鈭� 145 + 121}{18} \\ = 鈭� \frac{4}{3} \end{matrix}

Therefore, the solution of given system of equation is $(- \frac{4}{3}, \frac{3}{5}, - 11)$ .

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Most popular questions from this chapter

Use matrices $A$ , $B$ , $C$ and $D$ to find the following.

$A = [\begin{matrix} 5 & 7 \\ 鈭� 1 & 6 \\ 3 & 鈭� 9 \end{matrix}], 鈥夆赌� B = [\begin{matrix} 8 & 3 \\ 5 & 1 \\ 4 & 4 \end{matrix}], 鈥夆赌� C = [\begin{matrix} 0 & 4 \\ 鈭� 2 & 5 \\ 7 & 鈭� 1 \end{matrix}], 鈥夆赌� D = [\begin{matrix} 6 & 2 \\ 9 & 0 \\ 鈭� 3 & 0 \end{matrix}]$

25. $D - B$

State the dimensions of each matrix.

[\begin{matrix} 6 & 鈭� 1 & 5 \\ 鈭� 2 & 3 & 鈭� 4 \end{matrix}]

For exercises 26 and 27, use the advertisement shown at the right.

What are the dimensions of the matrix?

Find each product, if possible.10 $[\begin{matrix} 3 & 鈭� 1 \\ 2 & 5 \end{matrix}] . [\begin{matrix} 4 & 鈭� 1 & 鈭� 2 \\ 鈭� 3 & 5 & 4 \end{matrix}]$

Describe the conditions that must be met in order for two matrices to be considered equal.

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