This guided notes worksheet helps students master the elimination method for solving systems of linear equations with step-by-step instructions.

Problem Analysis:

The worksheet focuses on solving systems of linear equations using the elimination method. The goal is to fill in the blanks in the explanation and solve the given examples step by step.

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Step-by-Step Solution:

#### 1. Filling in the Blanks:
The elimination method involves systematically eliminating one of the variables to solve for the other. Let's fill in the blanks based on this understanding:

- Elimination is a method for solving systems of linear equations where you ________ one of the variables.
- Answer: Eliminate

- Steps:
1. Line up the equations so that the variables are in ________.
- Answer: Same order
2. Multiply one or both equations so that one of the ________ will cancel.
- Answer: Coefficients
3. Combine like ________ and ________ one of the variables.
- Answer: Terms, eliminate
4. Solve for the variable that did not ________.
- Answer: Cancel
5. Use the answer from step #4 and substitute into one of the original equations.
6. Solve for the other ________.
- Answer: Variable

- If everything cancels, the system has ________ solutions.
- Answer: Infinite
- If the variables cancel but the constants don’t, the system has ______ solution.
- Answer: No

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#### 2. Solving the Example:
The example provided is:
\[
2x + 3y = 10 \quad \text{(Equation 1)}
\]
\[
x + 2y = 4 \quad \text{(Equation 2)}
\]

##### Step 1: Line up the equations.
Both equations are already lined up with \(x\) and \(y\) in the same order.

##### Step 2: Multiply one or both equations to cancel a variable.
To eliminate \(x\), we can multiply Equation 2 by 2:
\[
2(x + 2y) = 2(4) \implies 2x + 4y = 8 \quad \text{(New Equation 2)}
\]

Now we have:
\[
2x + 3y = 10 \quad \text{(Equation 1)}
\]
\[
2x + 4y = 8 \quad \text{(New Equation 2)}
\]

##### Step 3: Combine like terms to eliminate one variable.
Subtract Equation 1 from New Equation 2:
\[
(2x + 4y) - (2x + 3y) = 8 - 10
\]
\[
2x + 4y - 2x - 3y = -2
\]
\[
y = -2
\]

##### Step 4: Solve for the remaining variable.
We found \(y = -2\).

##### Step 5: Substitute \(y = -2\) into one of the original equations.
Substitute \(y = -2\) into Equation 2:
\[
x + 2(-2) = 4
\]
\[
x - 4 = 4
\]
\[
x = 8
\]

##### Step 6: Solve for the other variable.
We already solved for \(x\) and found \(x = 8\).

##### Solution:
The solution to the system is:
\[
(x, y) = (8, -2)
\]

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#### 3. Solving More Examples:

##### Example 1:
\[
2x + y = 19 \quad \text{(Equation 1)}
\]
\[
x - y = 11 \quad \text{(Equation 2)}
\]

##### Step 1: Line up the equations.
Both equations are already lined up.

##### Step 2: Multiply one or both equations to cancel a variable.
To eliminate \(y\), add Equation 1 and Equation 2:
\[
(2x + y) + (x - y) = 19 + 11
\]
\[
2x + y + x - y = 30
\]
\[
3x = 30
\]
\[
x = 10
\]

##### Step 3: Solve for the remaining variable.
We found \(x = 10\).

##### Step 4: Substitute \(x = 10\) into one of the original equations.
Substitute \(x = 10\) into Equation 2:
\[
10 - y = 11
\]
\[
-y = 1
\]
\[
y = -1
\]

##### Solution:
The solution to the system is:
\[
(x, y) = (10, -1)
\]

##### Example 2:
\[
x + 2y = 5 \quad \text{(Equation 1)}
\]
\[
5x - y = 3 \quad \text{(Equation 2)}
\]

##### Step 1: Line up the equations.
Both equations are already lined up.

##### Step 2: Multiply one or both equations to cancel a variable.
To eliminate \(y\), multiply Equation 1 by 1 and Equation 2 by 2:
\[
x + 2y = 5 \quad \text{(Equation 1)}
\]
\[
2(5x - y) = 2(3) \implies 10x - 2y = 6 \quad \text{(New Equation 2)}
\]

Now we have:
\[
x + 2y = 5 \quad \text{(Equation 1)}
\]
\[
10x - 2y = 6 \quad \text{(New Equation 2)}
\]

Add Equation 1 and New Equation 2:
\[
(x + 2y) + (10x - 2y) = 5 + 6
\]
\[
x + 2y + 10x - 2y = 11
\]
\[
11x = 11
\]
\[
x = 1
\]

##### Step 3: Solve for the remaining variable.
We found \(x = 1\).

##### Step 4: Substitute \(x = 1\) into one of the original equations.
Substitute \(x = 1\) into Equation 1:
\[
1 + 2y = 5
\]
\[
2y = 4
\]
\[
y = 2
\]

##### Solution:
The solution to the system is:
\[
(x, y) = (1, 2)
\]

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Final Answers:

1. Example Solution:
\[
\boxed{(8, -2)}
\]

2. More Examples:
- Example 1: \(\boxed{(10, -1)}\)
- Example 2: \(\boxed{(1, 2)}\)

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This completes the solution. If you have any further questions, feel free to ask!