Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Explore thousands of questions and answers from a knowledgeable community of experts ready to help you find solutions. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
Let's solve the given systems of equations step-by-step using the specified methods.
### Question 1: Solving the System using the Method of Substitution
The system of equations is:
1. [tex]\(3x - 5y = 7\)[/tex]
2. [tex]\(2x + y = 9\)[/tex]
#### Step-by-Step Solution
1. Solve the second equation for [tex]\(y\)[/tex]:
[tex]\[ 2x + y = 9 \][/tex]
[tex]\[ y = 9 - 2x \][/tex]
2. Substitute [tex]\(y = 9 - 2x\)[/tex] into the first equation:
[tex]\[ 3x - 5(9 - 2x) = 7 \][/tex]
[tex]\[ 3x - 45 + 10x = 7 \][/tex]
[tex]\[ 13x - 45 = 7 \][/tex]
[tex]\[ 13x = 52 \][/tex]
[tex]\[ x = \frac{52}{13} \][/tex]
[tex]\[ x = 4 \][/tex]
3. Substitute [tex]\(x = 4\)[/tex] back into [tex]\(y = 9 - 2x\)[/tex]:
[tex]\[ y = 9 - 2(4) \][/tex]
[tex]\[ y = 9 - 8 \][/tex]
[tex]\[ y = 1 \][/tex]
The solution to the system using the Method of Substitution is:
[tex]\[ x = 4 \][/tex]
[tex]\[ y = 1 \][/tex]
### Question 2: Solving the System using the Elimination Method
The system of equations is:
1. [tex]\(2x + 3y = 18\)[/tex]
2. [tex]\(5x - y = 11\)[/tex]
#### Step-by-Step Solution
1. Multiply the second equation by 3 to align the coefficients of [tex]\(y\)[/tex]:
[tex]\[ 5x - y = 11 \][/tex]
[tex]\[ 3(5x - y) = 3(11) \][/tex]
[tex]\[ 15x - 3y = 33 \][/tex]
2. Add the equations to eliminate [tex]\(y\)[/tex]:
[tex]\[ (2x + 3y) + (15x - 3y) = 18 + 33 \][/tex]
[tex]\[ 2x + 3y + 15x - 3y = 51 \][/tex]
[tex]\[ 17x = 51 \][/tex]
[tex]\[ x = \frac{51}{17} \][/tex]
[tex]\[ x = 3 \][/tex]
3. Substitute [tex]\(x = 3\)[/tex] back into one of the original equations to solve for [tex]\(y\)[/tex]:
Using the first equation [tex]\(2x + 3y = 18\)[/tex]:
[tex]\[ 2(3) + 3y = 18 \][/tex]
[tex]\[ 6 + 3y = 18 \][/tex]
[tex]\[ 3y = 12 \][/tex]
[tex]\[ y = \frac{12}{3} \][/tex]
[tex]\[ y = 4 \][/tex]
The solution to the system using the Elimination Method is:
[tex]\[ x = 3 \][/tex]
[tex]\[ y = 4 \][/tex]
In summary, the solutions are:
1. Using the Method of Substitution: [tex]\( (x, y) = (4, 1) \)[/tex]
2. Using the Elimination Method: [tex]\( (x, y) = (3, 4) \)[/tex]
### Question 1: Solving the System using the Method of Substitution
The system of equations is:
1. [tex]\(3x - 5y = 7\)[/tex]
2. [tex]\(2x + y = 9\)[/tex]
#### Step-by-Step Solution
1. Solve the second equation for [tex]\(y\)[/tex]:
[tex]\[ 2x + y = 9 \][/tex]
[tex]\[ y = 9 - 2x \][/tex]
2. Substitute [tex]\(y = 9 - 2x\)[/tex] into the first equation:
[tex]\[ 3x - 5(9 - 2x) = 7 \][/tex]
[tex]\[ 3x - 45 + 10x = 7 \][/tex]
[tex]\[ 13x - 45 = 7 \][/tex]
[tex]\[ 13x = 52 \][/tex]
[tex]\[ x = \frac{52}{13} \][/tex]
[tex]\[ x = 4 \][/tex]
3. Substitute [tex]\(x = 4\)[/tex] back into [tex]\(y = 9 - 2x\)[/tex]:
[tex]\[ y = 9 - 2(4) \][/tex]
[tex]\[ y = 9 - 8 \][/tex]
[tex]\[ y = 1 \][/tex]
The solution to the system using the Method of Substitution is:
[tex]\[ x = 4 \][/tex]
[tex]\[ y = 1 \][/tex]
### Question 2: Solving the System using the Elimination Method
The system of equations is:
1. [tex]\(2x + 3y = 18\)[/tex]
2. [tex]\(5x - y = 11\)[/tex]
#### Step-by-Step Solution
1. Multiply the second equation by 3 to align the coefficients of [tex]\(y\)[/tex]:
[tex]\[ 5x - y = 11 \][/tex]
[tex]\[ 3(5x - y) = 3(11) \][/tex]
[tex]\[ 15x - 3y = 33 \][/tex]
2. Add the equations to eliminate [tex]\(y\)[/tex]:
[tex]\[ (2x + 3y) + (15x - 3y) = 18 + 33 \][/tex]
[tex]\[ 2x + 3y + 15x - 3y = 51 \][/tex]
[tex]\[ 17x = 51 \][/tex]
[tex]\[ x = \frac{51}{17} \][/tex]
[tex]\[ x = 3 \][/tex]
3. Substitute [tex]\(x = 3\)[/tex] back into one of the original equations to solve for [tex]\(y\)[/tex]:
Using the first equation [tex]\(2x + 3y = 18\)[/tex]:
[tex]\[ 2(3) + 3y = 18 \][/tex]
[tex]\[ 6 + 3y = 18 \][/tex]
[tex]\[ 3y = 12 \][/tex]
[tex]\[ y = \frac{12}{3} \][/tex]
[tex]\[ y = 4 \][/tex]
The solution to the system using the Elimination Method is:
[tex]\[ x = 3 \][/tex]
[tex]\[ y = 4 \][/tex]
In summary, the solutions are:
1. Using the Method of Substitution: [tex]\( (x, y) = (4, 1) \)[/tex]
2. Using the Elimination Method: [tex]\( (x, y) = (3, 4) \)[/tex]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.