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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]
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