the method where one equation is solved for a variable and substituted into another equation is called ______.
the method where equations are added or subtracted to eliminate a variable is called ______.
a set of two or more equations with the same variables is called a ______.
the point where the graphs of two equations intersect is called the ______.
when a system of equations has no solution, the lines are ______.
when a system of equations has infinite solutions, the equations are ______.

Question

Accepted Answer

### 1. First blank (method of solving equations)
# Brief Explanations:
The method where one equation is solved for a variable (like solving $ y = 2x + 1 $ from a system) and then substituted into another equation is the **substitution method**. This is a standard technique in solving systems of linear equations.
# Answer:
substitution method

### 2. Second blank (method of solving equations)
# Brief Explanations:
When we add or subtract equations (e.g., $ 2x + 3y = 5 $ and $ -2x + 5y = 7 $ are added to eliminate $ x $) to get rid of a variable, this is the **elimination method** (also called addition - subtraction method or linear combination method).
# Answer:
elimination method (or linear combination method)

### 3. Third blank (definition of a set of equations)
# Brief Explanations:
A collection of two or more equations that share the same variables (for example, $ \begin{cases}x + y = 3\2x - y = 1\end{cases} $) is called a **system of equations** (specifically, a system of linear equations if they are linear).
# Answer:
system of equations

### 4. Fourth blank (point of intersection of graphs)
# Brief Explanations:
The point where the graphs of two equations (like two lines in a coordinate plane) cross each other is the **solution** of the system of equations (since it satisfies both equations).
# Answer:
solution (of the system of equations)

### 5. Fifth blank (lines with no solution)
# Brief Explanations:
If a system of linear equations has no solution, it means the lines represented by the equations never meet. In a coordinate plane, such lines are **parallel** (they have the same slope but different y - intercepts).
# Answer:
parallel

### 6. Sixth blank (equations with infinite solutions)
# Brief Explanations:
When a system of equations has infinite solutions, the equations are **dependent** (and also coincident, meaning they represent the same line). For example, $ y = 2x + 1 $ and $ 2y = 4x + 2 $ are dependent equations as one can be derived from the other.
# Answer:
dependent (or coincident)

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QUESTION IMAGE

Question

Explanation:

1. First blank (method of solving equations)

Answer:

2. Second blank (method of solving equations)