Q25. The sum of 3 numbers is 12. The ... [FREE SOLUTION]

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Chapter 3: Q25. (page 143)

The sum of 3 numbers is 12. The first number is twice the sum of the second and third. The third number is 5 less than the first. Find the numbers.

Short Answer

Expert verified

The three numbers are $8, 1,$ and .3

Step by step solution

– First construct a system of equations corresponding to the given situation.

Let the three numbers be $x, y,$ and $z$ .

The sum of three numbers is 12, so, mathematically it can be written as $x + y + z = 12$ .

The first number is twice the second and third.

This can be mathematically written as $$ x = 2 \ l e f t (y + z \ r i g h t)$ $\ R i g h t a r r o w x - 2 y - 2 z = 0$ .

The third number is 5 less than the first.

This can be mathematically written as $z = x - 5 鈬� - x + z = - 5$ .

So, the system of equations will be:

$\begin{matrix} x + y + z = 12 \\ x 鈭� 2 y 鈭� 2 z = 0 \\ 鈭� x + z = 鈭� 5 \end{matrix}$

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

Add the $e q u a t i o n x + y + z = 12$ to the equation $- x + z = - 5$ .

$\begin{array}{l} \underline{\begin{array}{l} 鈥夆赌� x + y + z = 鈥夆赌夆€� 12 \\ 鈭� x + 鈥夆赌夆€夆赌� + z = 鈭� 鈥� 5 \end{array}} \\ 鈥夆赌夆€夆赌夆€夆赌夆€� y + 2 z = 鈥夆赌夆€� 7 \end{array}$

So, the resultant equation is $y + 2 z = 7$

Subtract the equation $x - 2 y - 2 z = 0$ from the equation $- x + z = - 5$

$\begin{array}{l} \underline{\begin{array}{l} 鈥夆赌� x 鈭� 2 y 鈭� 2 z = 鈥夆赌夆€� 0 \\ 鈭� x + 鈥夆赌夆€夆赌夆€� + 鈥夆赌夆€� z = 鈥� 鈭� 5 \end{array}} \\ 鈥夆赌夆€夆赌� 鈭� 2 y 鈭� 鈥夆赌夆€� z = 鈭� 5 \end{array}$

So, the resultant equation is $- 2 y - z = - 5$ .

– Solve the system of equations in two variables

Multiply $y + 2 z = 7$ by 2 and add the new resultant equation to $- 2 y - z = - 5 .$

$\begin{array}{l} \underline{\begin{array}{l} 鈥夆赌夆€� y + 2 z = 鈥夆赌夆€� 7 \\ 鈭� 2 y 鈭� 鈥� z = 鈭� 5 \end{array}} 鈥夆赌� \begin{matrix} multiplyby2 \end{matrix} 鈥夆赌夆€� \underline{\begin{array}{l} 鈥夆赌� 2 y + 4 z = 鈥夆赌� 14 \\ 鈭� 2 y 鈭� 鈥� z = 鈭� 鈥夆赌� 5 \end{array}} \\ 鈥夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌� 0 + 3 z = 鈥夆赌夆€� 9 \end{array}$

Solve $3 z = 9 f o r x$

$\begin{matrix} 3 z = 9 \\ \frac{3 z}{3} = \frac{9}{3} 鈥夆赌夆€夆赌夆€塪ividebothsidesby3 \\ z = 3 \end{matrix}$

– Find the values of yand z.

Substitute $z = 3$ in $y + 2 z = 7$ and find the value of $y$ .

$\begin{matrix} y + 2 z = 7 \\ y + 2 (3) = 7 鈥夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€塖ubstitute 3 for y \\ y + 6 = 7 鈥夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€塖implify \\ y = 1 鈥夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌塖ubtract 6 frombothsides \end{matrix}$

Substitute $z = 3$ in $- x + z = - 5$ and find the value of $x$ .

$\begin{matrix} 鈭� x + z = 鈭� 5 \\ 鈭� x + 3 = 鈭� 5 鈥夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€塻ubstitute 3 for z \\ 鈭� x = 鈭� 8 鈥夆赌夆€夆赌夆€夆赌夆€夆赌夆€夆赌夆€塖ubtract 3 鈥塮谤辞尘产辞迟丑蝉颈诲别蝉 \\ x = 8 Dividebothsidesby 鈭� 1 \end{matrix}$

Hence, the solution of the given system of equations is $(x, y, z) = (8, 1, 3)$ .

So, the three numbers are $8, 1,$ and $3$ .

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91影视

The sum of 3 numbers is 12. The first number is twice the sum of the second and third. The third number is 5 less than the first. Find the numbers.

Short Answer

Step by step solution

– First construct a system of equations corresponding to the given situation.

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

– Solve the system of equations in two variables

– Find the values of yand z.

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