At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
