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Sagot :
Sure! Let's go through the problem step-by-step and fill in the values for the expressions, and then identify the equivalent product and sum expressions.
### Step-by-Step Solution:
Step 1: Calculate the expressions for each set of values
We need to calculate three expressions for each given set of values of [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex]:
- [tex]\( 3(x + 2y + 4z) \)[/tex]
- [tex]\( 3x + 6y + 12z \)[/tex]
- [tex]\( 3x + 2y + 4z \)[/tex]
Given values:
1. [tex]\( x = 1 \)[/tex], [tex]\( y = 2 \)[/tex], [tex]\( z = 3 \)[/tex]
2. [tex]\( x = 10 \)[/tex], [tex]\( y = 20 \)[/tex], [tex]\( z = 30 \)[/tex]
3. [tex]\( x = 23 \)[/tex], [tex]\( y = 60 \)[/tex], [tex]\( z = 10 \)[/tex]
4. [tex]\( x = 14 \)[/tex], [tex]\( y = 0 \)[/tex], [tex]\( z = 1 \)[/tex]
5. [tex]\( x = 5 \)[/tex], [tex]\( y = 9 \)[/tex], [tex]\( z = 32 \)[/tex]
Using these values, let's calculate the expressions:
1. For [tex]\( x = 1 \)[/tex], [tex]\( y = 2 \)[/tex], [tex]\( z = 3 \)[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(1 + 2(2) + 4(3)) = 3(1 + 4 + 12) = 3 \times 17 = 51 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(1) + 6(2) + 12(3) = 3 + 12 + 36 = 51 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(1) + 2(2) + 4(3) = 3 + 4 + 12 = 19 \][/tex]
2. For [tex]\( x = 10 \)[/tex], [tex]\( y = 20 \)[/tex], [tex]\( z = 30 \)[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(10 + 2(20) + 4(30)) = 3(10 + 40 + 120) = 3 \times 170 = 510 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(10) + 6(20) + 12(30) = 30 + 120 + 360 = 510 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(10) + 2(20) + 4(30) = 30 + 40 + 120 = 190 \][/tex]
3. For [tex]\( x = 23 \)[/tex], [tex]\( y = 60 \)[/tex], [tex]\( z = 10 \)[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(23 + 2(60) + 4(10)) = 3(23 + 120 + 40) = 3 \times 183 = 549 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(23) + 6(60) + 12(10) = 69 + 360 + 120 = 549 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(23) + 2(60) + 4(10) = 69 + 120 + 40 = 229 \][/tex]
4. For [tex]\( x = 14 \)[/tex], [tex]\( y = 0 \)[/tex], [tex]\( z = 1 \)[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(14 + 2(0) + 4(1)) = 3(14 + 0 + 4) = 3 \times 18 = 54 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(14) + 6(0) + 12(1) = 42 + 0 + 12 = 54 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(14) + 2(0) + 4(1) = 42 + 0 + 4 = 46 \][/tex]
5. For [tex]\( x = 5 \)[/tex], [tex]\( y = 9 \)[/tex], [tex]\( z = 32 \)[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(5 + 2(9) + 4(32)) = 3(5 + 18 + 128) = 3 \times 151 = 453 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(5) + 6(9) + 12(32) = 15 + 54 + 384 = 453 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(5) + 2(9) + 4(32) = 15 + 18 + 128 = 161 \][/tex]
### Step 2: Fill in the Table
[tex]\[ \begin{array}{|c|c|c|c|} \hline & 3(x + 2y + 4z) & 3x + 6y + 12z & 3x + 2y + 4z \\ \hline x = 1, y = 2, z = 3 & 51 & 51 & 19 \\ \hline x = 10, y = 20, z = 30 & 510 & 510 & 190 \\ \hline x = 23, y = 60, z = 10 & 549 & 549 & 229 \\ \hline x = 14, y = 0, z = 1 & 54 & 54 & 46 \\ \hline x = 5, y = 9, z = 32 & 453 & 453 & 161 \\ \hline \end{array} \][/tex]
### Step 3: Identify Equivalent Expressions
The expressions [tex]\( 3(x + 2y + 4z) \)[/tex] and [tex]\( 3x + 6y + 12z \)[/tex] are equivalent, as shown by the calculations.
So the equivalent expressions are:
[tex]\[ 3(x + 2y + 4z) \text{ and } 3x + 6y + 12z \][/tex]
### Step-by-Step Solution:
Step 1: Calculate the expressions for each set of values
We need to calculate three expressions for each given set of values of [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex]:
- [tex]\( 3(x + 2y + 4z) \)[/tex]
- [tex]\( 3x + 6y + 12z \)[/tex]
- [tex]\( 3x + 2y + 4z \)[/tex]
Given values:
1. [tex]\( x = 1 \)[/tex], [tex]\( y = 2 \)[/tex], [tex]\( z = 3 \)[/tex]
2. [tex]\( x = 10 \)[/tex], [tex]\( y = 20 \)[/tex], [tex]\( z = 30 \)[/tex]
3. [tex]\( x = 23 \)[/tex], [tex]\( y = 60 \)[/tex], [tex]\( z = 10 \)[/tex]
4. [tex]\( x = 14 \)[/tex], [tex]\( y = 0 \)[/tex], [tex]\( z = 1 \)[/tex]
5. [tex]\( x = 5 \)[/tex], [tex]\( y = 9 \)[/tex], [tex]\( z = 32 \)[/tex]
Using these values, let's calculate the expressions:
1. For [tex]\( x = 1 \)[/tex], [tex]\( y = 2 \)[/tex], [tex]\( z = 3 \)[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(1 + 2(2) + 4(3)) = 3(1 + 4 + 12) = 3 \times 17 = 51 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(1) + 6(2) + 12(3) = 3 + 12 + 36 = 51 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(1) + 2(2) + 4(3) = 3 + 4 + 12 = 19 \][/tex]
2. For [tex]\( x = 10 \)[/tex], [tex]\( y = 20 \)[/tex], [tex]\( z = 30 \)[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(10 + 2(20) + 4(30)) = 3(10 + 40 + 120) = 3 \times 170 = 510 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(10) + 6(20) + 12(30) = 30 + 120 + 360 = 510 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(10) + 2(20) + 4(30) = 30 + 40 + 120 = 190 \][/tex]
3. For [tex]\( x = 23 \)[/tex], [tex]\( y = 60 \)[/tex], [tex]\( z = 10 \)[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(23 + 2(60) + 4(10)) = 3(23 + 120 + 40) = 3 \times 183 = 549 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(23) + 6(60) + 12(10) = 69 + 360 + 120 = 549 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(23) + 2(60) + 4(10) = 69 + 120 + 40 = 229 \][/tex]
4. For [tex]\( x = 14 \)[/tex], [tex]\( y = 0 \)[/tex], [tex]\( z = 1 \)[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(14 + 2(0) + 4(1)) = 3(14 + 0 + 4) = 3 \times 18 = 54 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(14) + 6(0) + 12(1) = 42 + 0 + 12 = 54 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(14) + 2(0) + 4(1) = 42 + 0 + 4 = 46 \][/tex]
5. For [tex]\( x = 5 \)[/tex], [tex]\( y = 9 \)[/tex], [tex]\( z = 32 \)[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(5 + 2(9) + 4(32)) = 3(5 + 18 + 128) = 3 \times 151 = 453 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(5) + 6(9) + 12(32) = 15 + 54 + 384 = 453 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(5) + 2(9) + 4(32) = 15 + 18 + 128 = 161 \][/tex]
### Step 2: Fill in the Table
[tex]\[ \begin{array}{|c|c|c|c|} \hline & 3(x + 2y + 4z) & 3x + 6y + 12z & 3x + 2y + 4z \\ \hline x = 1, y = 2, z = 3 & 51 & 51 & 19 \\ \hline x = 10, y = 20, z = 30 & 510 & 510 & 190 \\ \hline x = 23, y = 60, z = 10 & 549 & 549 & 229 \\ \hline x = 14, y = 0, z = 1 & 54 & 54 & 46 \\ \hline x = 5, y = 9, z = 32 & 453 & 453 & 161 \\ \hline \end{array} \][/tex]
### Step 3: Identify Equivalent Expressions
The expressions [tex]\( 3(x + 2y + 4z) \)[/tex] and [tex]\( 3x + 6y + 12z \)[/tex] are equivalent, as shown by the calculations.
So the equivalent expressions are:
[tex]\[ 3(x + 2y + 4z) \text{ and } 3x + 6y + 12z \][/tex]
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