2-2 2 Linear Equations Explained
Key Concepts of 2-2 2 Linear Equations
2-2 2 Linear Equations refer to a system of three linear equations with two variables each. This system can be solved using various methods such as substitution, elimination, and graphing. Understanding these methods is crucial for solving more complex problems in algebra and beyond.
1. System of Linear Equations
A system of linear equations consists of two or more equations that share the same variables. In the case of 2-2 2 Linear Equations, we have three equations with two variables each. For example:
\[ \begin{cases} 2x + 3y = 8 \\ 4x - y = 1 \\ x + 2y = 5 \end{cases} \]
2. Solving by Substitution
The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equations. This reduces the system to a simpler form.
Example:
Solve the system:
\[ \begin{cases} 2x + 3y = 8 \\ 4x - y = 1 \\ x + 2y = 5 \end{cases} \]
Step 1: Solve the second equation for \( y \):
\[ y = 4x - 1 \]
Step 2: Substitute \( y \) into the first and third equations:
\[ 2x + 3(4x - 1) = 8 \]
\[ x + 2(4x - 1) = 5 \]
Step 3: Simplify and solve for \( x \):
\[ 2x + 12x - 3 = 8 \]
\[ 14x - 3 = 8 \]
\[ 14x = 11 \]
\[ x = \frac{11}{14} \]
Step 4: Substitute \( x \) back into \( y = 4x - 1 \) to find \( y \):
\[ y = 4 \left(\frac{11}{14}\right) - 1 \]
\[ y = \frac{44}{14} - 1 \]
\[ y = \frac{30}{14} \]
\[ y = \frac{15}{7} \]
3. Solving by Elimination
The elimination method involves adding or subtracting equations to eliminate one of the variables. This method is particularly useful when the coefficients of one of the variables are the same or opposites.
Example:
Solve the system:
\[ \begin{cases} 2x + 3y = 8 \\ 4x - y = 1 \\ x + 2y = 5 \end{cases} \]
Step 1: Multiply the second equation by 2 to align the coefficients of \( y \):
\[ 8x - 2y = 2 \]
Step 2: Add the first and modified second equations:
\[ 2x + 3y + 8x - 2y = 8 + 2 \]
\[ 10x + y = 10 \]
Step 3: Solve for \( y \):
\[ y = 10 - 10x \]
Step 4: Substitute \( y \) into the third equation:
\[ x + 2(10 - 10x) = 5 \]
\[ x + 20 - 20x = 5 \]
\[ -19x + 20 = 5 \]
\[ -19x = -15 \]
\[ x = \frac{15}{19} \]
Step 5: Substitute \( x \) back into \( y = 10 - 10x \) to find \( y \):
\[ y = 10 - 10 \left(\frac{15}{19}\right) \]
\[ y = 10 - \frac{150}{19} \]
\[ y = \frac{190}{19} - \frac{150}{19} \]
\[ y = \frac{40}{19} \]
4. Solving by Graphing
Graphing involves plotting the lines of the equations on a coordinate plane and finding the point where they intersect. This method is useful for visualizing the solution but may not be as precise as algebraic methods.
Example:
Graph the system:
\[ \begin{cases} 2x + 3y = 8 \\ 4x - y = 1 \\ x + 2y = 5 \end{cases} \]
Plot each equation on the coordinate plane and find the common intersection point. The coordinates of this point represent the solution to the system.
Practical Applications
Understanding 2-2 2 Linear Equations is essential for solving real-world problems, such as optimizing resource allocation, balancing chemical equations, and analyzing economic models. These skills are foundational for more advanced topics in mathematics and science.