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Chapter 3: Q7. (page 142)

Solve each system of equations.
7.
$\begin{array}{l} 2 r + 3 s 鈭� 4 t = 20 \\ 4 r 鈭� s + 5 t = 13 \\ 3 r + 2 s + 4 t = 15 \end{array}$

Short Answer

Expert verified

The solution is $r = 5, s = - 2, t = - 1 .$

Step by step solution

01

Step-1 –Using elimination to obtain two equations with two variables

Given system of equations are

$\begin{array}{l} 2 r + 3 s 鈭� 4 t = 20 \\ 4 r 鈭� s + 5 t = 13 \\ 3 r + 2 s + 4 t = 15 \end{array}$

Multiplying second equation by $3$ and adding to the first equation, we get

$\begin{array}{l} 2 r + 12 r 鈭� 4 t + 15 t = 20 + 39 \\ 14 r + 11 t = 59 \end{array}$

Again multiplying second equation by $2$ and adding to the third equation, we get

$\begin{array}{l} 8 r + 3 r + 10 t + 4 t = 26 + 15 \\ 11 r + 14 t = 41 \end{array}$

02

Step-2 –Solving the system of two equations with two variables

System of two equations with two variables are

$\begin{array}{l} 114 r + 11 t = 59 \\ 11 r + 14 t = 41 \end{array}$

Multiplying the first equation by $11$ and second by $14$ and subtracting them we get

$\begin{matrix} 121 t - 196 t = 649 - 574 \\ - 75 t = 75 \\ t = - 1 \end{matrix}$

Putting the value of $t$ in equation $14 r + 11 t = 59$ we get

$\begin{array}{l} 14 r 鈭� 11 = 59 \\ 14 r = 59 + 11 \\ 14 r = 70 \\ r = 5 \end{array}$

03

Step-3 –Finding the value of s using the value of r and t

Putting the value of $r$ and $t$ in the equation $\ [4 r - s + 5 t \] \ [4 r - s + 5 t \]$ we get,

$\begin{array}{l} 4 (5) 鈭� s + 5 (鈭� 1) = 13 \\ 20 鈭� s 鈭� 5 = 13 \\ 鈭� s = 13 鈭� 15 \\ 鈭� s = 鈭� 2 \\ s 鈭� 2 \end{array}$

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Most popular questions from this chapter

Solve each system of equations by using either substitution or elimination $\begin{array}{l} 9.5 m + n = 10 \\ 4 m + n = 4 \end{array}$

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent.

$\begin{matrix} y 鈭� x = 5 \\ 2 y 鈭� 2 x = 8 \end{matrix}$

Solve each system of equations by graphing.

19. $\begin{matrix} 2 x + 3 y = 7 \\ 2 x 鈭� 3 y = 7 \end{matrix}$

Simplify each expression.

$(3 y - 11) - (6 y + 12)$

Asportinggoodsmanufacturermakesa\(5profitonsoccerballsanda\)4profitonvolleyballs.Cuttingrequires2hourstomake75soccerballsand3hourstomake60volleyballs.Sewingneeds3hourstomake75soccerballsand2hourstomake60volleyballs.Cuttinghas500hoursavailable,andsewinghas450hoursavailable.

What is the maximum profit the company can make from these two products?

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