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Learn solving linear equation systems with substitution and elimination methods. Learn real-world applications with worked examples and visual guides.

---

## 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:

```math
a_1x_1 + a_2x_2 + ... + a_nx_n = b
```

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

Visible text: Where are coefficients, are variables, and 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:

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

```math
\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}
```

**Example**:

```math
\begin{cases} 2x + 3y = 21 & \ldots (1) \\ x + y = 10 & \ldots (2) \end{cases}
```

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

Visible text: The solution to this system is the pair of values that satisfies both equations.

### Three-Variable Linear Equation Systems

For three variables, we need at least three equations:

```math
\begin{cases} a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3 \end{cases}
```

**Example**:

```math
\begin{cases} x + 2y + 3z = 27 \\ x + y + z = 16 \\ x = 6 \end{cases}
```

## Solving Linear Equation Systems

### Substitution Method

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

```math
\begin{cases} 2x + 3y = 21 & \ldots (1) \\ x + y = 10 & \ldots (2) \end{cases}
```

**Step** $$1$$: Express one variable from the simpler equation.

Visible text: **Step** : Express one variable from the simpler equation.

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

Visible text: From equation (): , we express in terms of :

```math
x = 10 - y \ldots (3)
```

**Step** $$2$$: Substitute into the other equation.

Visible text: **Step** : Substitute into the other equation.

Insert equation ($$3$$) into equation ($$1$$):

Visible text: Insert equation () into equation ():

```math
2(10 - y) + 3y = 21
```

**Step** $$3$$: Solve the resulting equation.

Visible text: **Step** : Solve the resulting equation.

Component: MathContainer
Children:

```math
20 - 2y + 3y = 21
```

```math
20 + y = 21
```

```math
y = 1 \ldots (4)
```

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

Visible text: **Step** : Back-substitute to find the other variable.

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

Visible text: Substitute the value from equation () into equation ():

```math
x = 10 - y = 10 - 1 = 9
```

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

Visible text: Therefore, the solution is and .

### Elimination Method

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

```math
\begin{cases} 2x + 3y = 21 & \ldots (1) \\ x + y = 10 & \ldots (2) \end{cases}
```

**Step** $$1$$: Match the coefficients of one variable.

Visible text: **Step** : Match the coefficients of one variable.

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

Visible text: Multiply equation () by to match the coefficient of :

```math
\begin{cases} 2x + 3y = 21 & \ldots (1) \\ 2x + 2y = 20 & \ldots (3) \end{cases}
```

**Step** $$2$$: Eliminate the variable by subtracting the equations.

Visible text: **Step** : Eliminate the variable by subtracting the equations.

Component: MathContainer
Children:

```math
(1) - (3): (2x + 3y) - (2x + 2y) = 21 - 20
```

```math
y = 1 \ldots (4)
```

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

Visible text: **Step** : Use the value of to find .

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

Visible text: Substitute the value from equation () into equation ():

Component: MathContainer
Children:

```math
x + 1 = 10
```

```math
x = 9
```

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

Visible text: Therefore, the solution is and .

Verify:

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

Visible text: - Equation (): ✓
- Equation (): ✓

## 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).

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

Let's define:

- $$a$$ = number of $$1\text{-point}$$ shots
- $$b$$ = number of $$2\text{-point}$$ shots
- $$c$$ = number of $$3\text{-point}$$ shots

Visible text: - = number of shots
- = number of shots
- = number of shots

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

Visible text: If Wijaya scored points, made shots total with of them being free throws, then:

```math
\begin{cases} a + 2b + 3c = 27 & \text{(total points)} \\ a + b + c = 16 & \text{(total shots)} \\ a = 6 & \text{(free throws)} \end{cases}
```

Substituting $$a = 6$$ into the second equation:

Visible text: Substituting into the second equation:

Component: MathContainer
Children:

```math
6 + b + c = 16
```

```math
b + c = 10
```

Substituting into the first equation:

Component: MathContainer
Children:

```math
6 + 2b + 3c = 27
```

```math
2b + 3c = 21
```

From these two equations:

```math
\begin{cases} 2b + 3c = 21 \\ b + c = 10 \end{cases}
```

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

Visible text: Using elimination or substitution, we get and .

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

Visible text: Therefore, Wijaya made free throws, two-point shots, and 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

Component: ContentStack
Children:
Component: LineEquation
Props:
- title: Linear Equation System with One Solution
- description: Two lines intersecting at a single point.
- data: [
{
points: [
{ x: -2, y: 8, z: 0 },
{ x: 3.5, y: 4.5, z: 0 },
{ x: 9, y: 1, z: 0 },
],
labels: [{ text: "2x + 3y = 21", offset: [-1, -1, 0] }],
color: getColor("ORANGE"),
},
{
points: [
{ x: -5, y: 15, z: 0 },
{ x: 5, y: 5, z: 0 },
{ x: 15, y: -5, z: 0 },
],
labels: [{ text: "x + y = 10", offset: [1, 1, 0] }],
color: getColor("PURPLE"),
},
]

Component: LineEquation
Props:
- title: Linear Equation System with No Solution
- description: Two parallel lines (no intersection).
- data: [
{
points: [
{ x: -3, y: 1, z: 0 },
{ x: 0, y: 4, z: 0 },
{ x: 3, y: 7, z: 0 },
],
labels: [{ text: "3x - 2y = -1", offset: [-2, 0, 0] }],
color: getColor("PINK"),
},
{
points: [
{ x: -2, y: -1, z: 0 },
{ x: 1, y: 2, z: 0 },
{ x: 4, y: 5, z: 0 },
],
labels: [{ text: "6x - 4y = 2", offset: [2, 0, 0] }],
color: getColor("YELLOW"),
},
]

Component: LineEquation
Props:
- title: Linear Equation System with Infinitely Many Solutions
- description: Two coincident lines (identical lines).
- data: [
{
points: [
{ x: -2, y: 1, z: 0 },
{ x: 0, y: 0, z: 0 },
{ x: 2, y: -1, z: 0 },
],
labels: [{ text: "2x + 3y = 1", offset: [-2, -1, 0] }],
color: getColor("VIOLET"),
},
{
points: [
{ x: -2, y: 1, z: 0 },
{ x: 0, y: 0, z: 0 },
{ x: 2, y: -1, z: 0 },
],
labels: [{ text: "4x + 6y = 2", offset: [2, 1, 0] }],
color: getColor("CYAN"),
},
]
- cameraPosition: [6, 4, 6]