Q21. Solve each system of equations聽... [FREE SOLUTION]

Chapter 3: Q21. (page 143)

Solve each system of equations
$\begin{array}{l} 6 x + 2 y + 4 z = 2 \\ 3 x + 4 y 鈭� 8 z = 鈭� 3 \\ 鈭� 3 x 鈭� 6 y + 12 z = 5 \end{array}$

Short Answer

Expert verified

The solution set of the given system of equations is $(\frac{1}{3}, - \frac{1}{2}, \frac{1}{4})$ .

Step by step solution

– Use the elimination method to get the system of equations in two variables.

Multiply the equation $6 x + 2 y + 4 z = 2$ by $2$ and add the new resultant equation to the equation $3 x + 4 y - 8 z = - 3$ .

localid="1647997506005" role="math" $\begin{array}{l} \underset{}{\begin{array}{l} 6 x + 2 y + 4 z = 鈥夆赌夆赌� 2 \\ 3 x + 4 y 鈭� 8 z = 鈭� 3 \end{array}} 鈥夆赌夆赌夆€� \begin{matrix} multiplyby2 \end{matrix} 鈥夆赌� \underset{}{\begin{array}{l} 12 x + 4 y + 8 z = 鈥夆赌� 4 \\ 鈥� 3 x + 4 y 鈭� 8 z = 鈭� 3 \end{array}} \\ 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌� 15 x + 8 y + 鈥夆赌� 0 鈥� = 鈥夆赌夆赌夆€� 1 \end{array}$

So, the resultant equation is $15 x + 8 y = 1$

Multiply the equation $3 x + 4 y - 8 z = - 3$ by $3$ and multiply the equation $- 3 x - 6 y + 12 z = 5$ by $2$ .

Then add the two new resultant equations.

$\begin{array}{l} \underline{\begin{array}{l} 鈥夆赌� 3 x + 4 y 鈭� 鈥夆赌� 8 z = 鈭� 3 \\ 鈭� 3 x 鈭� 6 y + 12 z = 鈥夆赌� 5 \end{array}} 鈥夆赌夆赌夆€� \begin{matrix} multiplyby3 鈫� \\ multiplyby2 鈫� \end{matrix} 鈥夆赌� \underline{\begin{array}{l} 鈥� 9 x + 12 y 鈭� 24 z = 鈭� 9 \\ 鈭� 6 x 鈭� 12 y + 24 z = 10 \end{array}} \\ 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌� 3 x 鈭� 鈥夆赌夆赌� 0 鈥� + 鈥夆赌夆赌夆€� 0 鈥� = 1 \end{array}$

So, the resultant equation is $3 x = 1$ .

– Use the elimination method to solve the system of two equations.

Solve $3 x = 1$ for $x$ :

$\begin{array}{l} 3 x = 1 \\ \frac{3 x}{3} = \frac{1}{3} 鈥夆赌夆赌夆€夆赌夆赌塂ividebothsidesby3 \\ x = \frac{1}{3} \end{array}$

– Find the values of yand z.

Substitute $x = \ f r a c {1} {3} $ i n 15 x + 8 y = 1$ and find the value of $y$ .

$\begin{array}{l} 15 x + 8 y = 1 \\ 15 (\frac{1}{3}) + 8 y = 1 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€塖ubstitute \frac{1}{3} for x \\ 5 + 8 y = 1 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€塖implify \\ 8 y = 鈭� 4 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌塖ubtract5frombothsides \\ y = 鈭� \frac{1}{2} 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€塂ividebothsidesby 8 \end{array}$

Substitute localid="1647997601231" $x = \frac{1}{3}$ ,localid="1647997625663" $y = - \frac{1}{2} i n 3 x + 4 y - 8 z = - 3$ and find the value of $z$

$\begin{array}{l} 3 x + 4 y 鈭� 8 z = 鈭� 3 \\ 3 (\frac{1}{3}) + 4 (鈭� \frac{1}{2}) 鈭� 8 z = 鈭� 3 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌塻ubstitute \frac{1}{3} for x, 鈭� \frac{1}{2} for y \\ 1 鈭� 2 鈭� 8 z = 鈭� 3 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌塻implify \\ 鈭� 1 鈭� 8 z = 鈭� 3 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌塻implify \\ 鈭� 8 z = 鈭� 2 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌塧dd1onbothsides \\ z = \frac{1}{4} 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌塂ividebothsidesby 鈭� 8 \end{array}$

Hence, the solution of the given system of equations is $(x, y, z) = (\frac{1}{3}, - \frac{1}{2}, \frac{1}{4})$ .

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91影视

Solve each system of equations
$\begin{array}{l} 6 x + 2 y + 4 z = 2 \\ 3 x + 4 y 鈭� 8 z = 鈭� 3 \\ 鈭� 3 x 鈭� 6 y + 12 z = 5 \end{array}$

Short Answer

Step by step solution

– Use the elimination method to get the system of equations in two variables.

– Use the elimination method to solve the system of two equations.

– Find the values of yand z.

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91影视

Solve each system of equations6x+2y+4z=23x+4y鈭�8z=鈭�3鈭�3x鈭�6y+12z=5

Short Answer

Step by step solution

– Use the elimination method to get the system of equations in two variables.

– Use the elimination method to solve the system of two equations.

– Find the values of yand z.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Pure Maths

Statistics

Applied Mathematics

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Solve each system of equations
$\begin{array}{l} 6 x + 2 y + 4 z = 2 \\ 3 x + 4 y 鈭� 8 z = 鈭� 3 \\ 鈭� 3 x 鈭� 6 y + 12 z = 5 \end{array}$