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Problem Statement:
There are various methods used to solve systems of linear equations in two variables. Here, you will be asked to solve systems using two methods. An example of both methods is provided for you to base your work on.

Questions:
1. Solve the following system using the Method of Substitution.
[tex]
\begin{array}{l}
3x - 5y = 7 \\
2x + y = 9
\end{array}
[/tex]

2. Solve the following system using the Elimination Method.
[tex]
\begin{array}{l}
2x + 3y = 18 \\
5x - y = 11
\end{array}
[/tex]

Sagot :

GinnyAnswer
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]
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