Linear Equation Systems

Understanding Linear Equation Systems

Imagine you're making a cake recipe. You know the cake requires eggs and flour in specific amounts. However, you only know the total ingredients and their ratios. This is similar to a linear equation system - we're looking for unknown values based on related information.

What Is a Linear Equation System?

A linear equation system is a collection of two or more linear equations that must be satisfied simultaneously. Each linear equation has the form:

Where $a_1, a_2, ..., a_n$ are coefficients, $x_1, x_2, ..., x_n$ are variables, and $b$ is a constant.

Two-Variable Linear Equation Systems

A two-variable linear equation system consists of two equations with two variables (usually $x$ and $y$). The general form is:

Example:

The solution to this system is the pair of values $(x,y)$ that satisfies both equations.

Three-Variable Linear Equation Systems

For three variables, we need at least three equations:

Example:

Solving Linear Equation Systems

Substitution Method

The substitution method works by replacing one variable with another. Let's solve the following example:

Step 1: Express one variable from the simpler equation.

From equation (2): $x + y = 10$, we express $x$ in terms of $y$:

Step 2: Substitute into the other equation.

Insert equation (3) into equation (1):

Step 3: Solve the resulting equation.

Step 4: Back-substitute to find the other variable.

Substitute the value $y = 1$ from equation (4) into equation (3):

Therefore, the solution is $x = 9$ and $y = 1$.

Elimination Method

The elimination method works by eliminating one variable. Let's solve the same example:

Step 1: Match the coefficients of one variable.

Multiply equation (2) by 2 to match the coefficient of $x$:

Step 2: Eliminate the variable by subtracting the equations.

Step 3: Use the value of $y$ to find $x$.

Substitute the value $y = 1$ from equation (4) into equation (2):

Therefore, the solution is $x = 9$ and $y = 1$.

Verify:

• Equation (1): $2(9) + 3(1) = 18 + 3 = 21$ ✓
• Equation (2): $9 + 1 = 10$ ✓

Real-Life Applications

Mathematical Modeling

Mathematical modeling is the process of converting real-world problems into mathematical form. For linear equation systems, we:

1. Identify the variables to use
2. Create a mathematical model based on the available information
3. Check if the model is a linear equation system
4. Solve the model using an appropriate method
5. Interpret the solution in the context of the original problem

Basketball Scoring

In basketball, there are three types of shots with different point values: free throws (1 point), two-point shots (2 points), and three-point shots (3 points).

Let's define:

• $a$ = number of 1-point shots
• $b$ = number of 2-point shots
• $c$ = number of 3-point shots

If Wijaya scored 27 points, made 16 shots total with 6 of them being free throws, then:

Substituting $a = 6$ into the second equation:

Substituting into the first equation:

From these two equations:

Using elimination or substitution, we get $b = 9$ and $c = 1$.

Therefore, Wijaya made 6 free throws, 9 two-point shots, and 1 three-point shot.

Interpreting Solutions

Linear equation systems have three possible solution types:

1. Exactly one solution: When the lines intersect at a single point (or planes intersect at a single point)
2. No solution: When the lines are parallel (or planes do not intersect)
3. Infinitely many solutions: When the lines coincide (or planes intersect along a line or plane)

In three dimensions (three variables), a linear equation is represented as a plane. The intersection of two planes forms a line, and the intersection of three planes can form a point.

Visualizing Linear Equation Systems