Answer :
To find the solution to the given system of linear equations, we can solve it step-by-step using the substitution or elimination method. Let's use the following system:
[tex]\[ \left\{\begin{array}{c} p+q+r=32 \\ p-r=4 \\ 2p+q=36 \end{array}\right. \][/tex]
### Step 1: Solve for [tex]\( r \)[/tex] in terms of [tex]\( p \)[/tex] from the second equation.
From the second equation [tex]\( p - r = 4 \)[/tex]:
[tex]\[ r = p - 4 \][/tex]
### Step 2: Substitute [tex]\( r = p - 4 \)[/tex] into the first and third equations.
Substituting [tex]\( r = p - 4 \)[/tex] into the first equation [tex]\( p + q + r = 32 \)[/tex]:
[tex]\[ p + q + (p - 4) = 32 \][/tex]
[tex]\[ 2p + q - 4 = 32 \][/tex]
[tex]\[ 2p + q = 36 \][/tex]
Notice that this equation is the same as the third equation [tex]\( 2p + q = 36 \)[/tex]. This means that our equations are consistent, and we move to the next step.
### Step 3: Solve for [tex]\( q \)[/tex] in terms of [tex]\( p \)[/tex] from [tex]\( 2p + q = 36 \)[/tex].
From [tex]\( 2p + q = 36 \)[/tex]:
[tex]\[ q = 36 - 2p \][/tex]
### Step 4: Substitute [tex]\( q = 36 - 2p \)[/tex] and [tex]\( r = p - 4 \)[/tex] back together to ensure they meet the first equation.
We substitute back into the equation to check:
[tex]\[ p + (36 - 2p) + (p - 4) = 32 \\ p + 36 - 2p + p - 4 = 32 \\ 36 - 4 = 32 \\ 32 = 32 \][/tex]
This is consistent, so we have the correct expressions.
### Step 5: Determine the values of [tex]\( p \)[/tex], [tex]\( q \)[/tex], and [tex]\( r \)[/tex].
Let's find [tex]\( p \)[/tex]:
We have the expressions:
[tex]\[ 2p + q = 36 \][/tex]
[tex]\[ q = 36 - 2p \][/tex]
We need to match these to the list of possible solutions:
A. [tex]\((16, 8, 8)\)[/tex]
B. [tex]\((14, 10, 8)\)[/tex]
C. [tex]\((14, 8, 10)\)[/tex]
D. [tex]\((10, 12, 10)\)[/tex]
E. [tex]\((10, 15, 10)\)[/tex]
We can systematically check these options:
1. Option A: [tex]\( p = 16 \)[/tex], [tex]\( q = 8 \)[/tex], [tex]\( r = 8 \)[/tex]
[tex]\[ p + q + r = 16 + 8 + 8 = 32 \quad \text{(true)} \][/tex]
[tex]\[ p - r = 16 - 8 = 8 \quad \text{(false)} \][/tex]
[tex]\[ 2p + q = 2(16) + 8 = 32 + 8 = 40 \quad \text{(false)} \][/tex]
2. Option B: [tex]\( p = 14 \)[/tex], [tex]\( q = 10 \)[/tex], [tex]\( r = 8 \)[/tex]
[tex]\[ p + q + r = 14 + 10 + 8 = 32 \quad \text{(true)} \][/tex]
[tex]\[ p - r = 14 - 8 = 6 \quad \text{(false)} \][/tex]
[tex]\[ 2p + q = 2(14) + 10 = 28 + 10 = 38 \quad \text{(false)} \][/tex]
3. Option C: [tex]\( p = 14 \)[/tex], [tex]\( q = 8 \)[/tex], [tex]\( r = 10 \)[/tex]
[tex]\[ p + q + r = 14 + 8 + 10 = 32 \quad \text{(true)} \][/tex]
[tex]\[ p - r = 14 - 10 = 4 \quad \text{(true)} \][/tex]
[tex]\[ 2p + q = 2(14) + 8 = 28 + 8 = 36 \quad \text{(true)} \][/tex]
Thus, the correct answer is:
C. [tex]\((14, 8, 10)\)[/tex]
[tex]\[ \left\{\begin{array}{c} p+q+r=32 \\ p-r=4 \\ 2p+q=36 \end{array}\right. \][/tex]
### Step 1: Solve for [tex]\( r \)[/tex] in terms of [tex]\( p \)[/tex] from the second equation.
From the second equation [tex]\( p - r = 4 \)[/tex]:
[tex]\[ r = p - 4 \][/tex]
### Step 2: Substitute [tex]\( r = p - 4 \)[/tex] into the first and third equations.
Substituting [tex]\( r = p - 4 \)[/tex] into the first equation [tex]\( p + q + r = 32 \)[/tex]:
[tex]\[ p + q + (p - 4) = 32 \][/tex]
[tex]\[ 2p + q - 4 = 32 \][/tex]
[tex]\[ 2p + q = 36 \][/tex]
Notice that this equation is the same as the third equation [tex]\( 2p + q = 36 \)[/tex]. This means that our equations are consistent, and we move to the next step.
### Step 3: Solve for [tex]\( q \)[/tex] in terms of [tex]\( p \)[/tex] from [tex]\( 2p + q = 36 \)[/tex].
From [tex]\( 2p + q = 36 \)[/tex]:
[tex]\[ q = 36 - 2p \][/tex]
### Step 4: Substitute [tex]\( q = 36 - 2p \)[/tex] and [tex]\( r = p - 4 \)[/tex] back together to ensure they meet the first equation.
We substitute back into the equation to check:
[tex]\[ p + (36 - 2p) + (p - 4) = 32 \\ p + 36 - 2p + p - 4 = 32 \\ 36 - 4 = 32 \\ 32 = 32 \][/tex]
This is consistent, so we have the correct expressions.
### Step 5: Determine the values of [tex]\( p \)[/tex], [tex]\( q \)[/tex], and [tex]\( r \)[/tex].
Let's find [tex]\( p \)[/tex]:
We have the expressions:
[tex]\[ 2p + q = 36 \][/tex]
[tex]\[ q = 36 - 2p \][/tex]
We need to match these to the list of possible solutions:
A. [tex]\((16, 8, 8)\)[/tex]
B. [tex]\((14, 10, 8)\)[/tex]
C. [tex]\((14, 8, 10)\)[/tex]
D. [tex]\((10, 12, 10)\)[/tex]
E. [tex]\((10, 15, 10)\)[/tex]
We can systematically check these options:
1. Option A: [tex]\( p = 16 \)[/tex], [tex]\( q = 8 \)[/tex], [tex]\( r = 8 \)[/tex]
[tex]\[ p + q + r = 16 + 8 + 8 = 32 \quad \text{(true)} \][/tex]
[tex]\[ p - r = 16 - 8 = 8 \quad \text{(false)} \][/tex]
[tex]\[ 2p + q = 2(16) + 8 = 32 + 8 = 40 \quad \text{(false)} \][/tex]
2. Option B: [tex]\( p = 14 \)[/tex], [tex]\( q = 10 \)[/tex], [tex]\( r = 8 \)[/tex]
[tex]\[ p + q + r = 14 + 10 + 8 = 32 \quad \text{(true)} \][/tex]
[tex]\[ p - r = 14 - 8 = 6 \quad \text{(false)} \][/tex]
[tex]\[ 2p + q = 2(14) + 10 = 28 + 10 = 38 \quad \text{(false)} \][/tex]
3. Option C: [tex]\( p = 14 \)[/tex], [tex]\( q = 8 \)[/tex], [tex]\( r = 10 \)[/tex]
[tex]\[ p + q + r = 14 + 8 + 10 = 32 \quad \text{(true)} \][/tex]
[tex]\[ p - r = 14 - 10 = 4 \quad \text{(true)} \][/tex]
[tex]\[ 2p + q = 2(14) + 8 = 28 + 8 = 36 \quad \text{(true)} \][/tex]
Thus, the correct answer is:
C. [tex]\((14, 8, 10)\)[/tex]
