# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/10/mathematics/linear-equation-inequality/system-linear-equation
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/linear-equation-inequality/system-linear-equation/en.mdx

Output docs content for large language models.

---

import { LineEquation } from "@repo/design-system/components/contents/line-equation";
import { getColor } from "@repo/design-system/lib/color";

export const metadata = {
  title: "Linear Equation Systems",
  description: "Master solving linear equation systems with substitution and elimination methods. Learn real-world applications with step-by-step examples and visual guides.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/18/2025",
  subject: "Systems of Linear Equations and Inequalities",
};

## 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:

<BlockMath math="a_1x_1 + a_2x_2 + ... + a_nx_n = b" />

Where <InlineMath math="a_1, a_2, ..., a_n" /> are coefficients, <InlineMath math="x_1, x_2, ..., x_n" /> are variables, and <InlineMath math="b" /> is a constant.

### Two-Variable Linear Equation Systems

A two-variable linear equation system consists of two equations with two variables (usually <InlineMath math="x" /> and <InlineMath math="y" />). The general form is:

<BlockMath math="\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}" />

**Example**:

<BlockMath 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 <InlineMath math="(x,y)" /> that satisfies both equations.

### Three-Variable Linear Equation Systems

For three variables, we need at least three equations:

<BlockMath 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**:

<BlockMath 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:

<BlockMath math="\begin{cases} 2x + 3y = 21 & \ldots (1) \\ x + y = 10 & \ldots (2) \end{cases}" />

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

From equation (2): <InlineMath math="x + y = 10" />, we express <InlineMath math="x" /> in terms of <InlineMath math="y" />:

<BlockMath math="x = 10 - y \ldots (3)" />

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

Insert equation (3) into equation (1):

<BlockMath math="2(10 - y) + 3y = 21" />

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

<div className="flex flex-col gap-4">
  <BlockMath math="20 - 2y + 3y = 21" />
  <BlockMath math="20 + y = 21" />
  <BlockMath math="y = 1 \ldots (4)" />
</div>

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

Substitute the value <InlineMath math="y = 1" /> from equation (4) into equation (3):

<BlockMath math="x = 10 - y = 10 - 1 = 9" />

Therefore, the solution is <InlineMath math="x = 9" /> and <InlineMath math="y = 1" />.

### Elimination Method

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

<BlockMath math="\begin{cases} 2x + 3y = 21 & \ldots (1) \\ x + y = 10 & \ldots (2) \end{cases}" />

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

Multiply equation (2) by 2 to match the coefficient of <InlineMath math="x" />:

<BlockMath math="\begin{cases} 2x + 3y = 21 & \ldots (1) \\ 2x + 2y = 20 & \ldots (3) \end{cases}" />

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

<div className="flex flex-col gap-4">
  <BlockMath math="(1) - (3): (2x + 3y) - (2x + 2y) = 21 - 20" />
  <BlockMath math="y = 1 \ldots (4)" />
</div>

**Step 3**: Use the value of <InlineMath math="y" /> to find <InlineMath math="x" />.

Substitute the value <InlineMath math="y = 1" /> from equation (4) into equation (2):

<div className="flex flex-col gap-4">
  <BlockMath math="x + 1 = 10" />
  <BlockMath math="x = 9" />
</div>

Therefore, the solution is <InlineMath math="x = 9" /> and <InlineMath math="y = 1" />.

Verify:

- Equation (1): <InlineMath math="2(9) + 3(1) = 18 + 3 = 21" /> ✓
- Equation (2): <InlineMath math="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:

- <InlineMath math="a" /> = number of 1-point shots
- <InlineMath math="b" /> = number of 2-point shots
- <InlineMath math="c" /> = number of 3-point shots

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

<BlockMath 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 <InlineMath math="a = 6" /> into the second equation:

<div className="flex flex-col gap-4">
  <BlockMath math="6 + b + c = 16" />
  <BlockMath math="b + c = 10" />
</div>

Substituting into the first equation:

<div className="flex flex-col gap-4">
  <BlockMath math="6 + 2b + 3c = 27" />
  <BlockMath math="2b + 3c = 21" />
</div>

From these two equations:

<BlockMath math="\begin{cases} 2b + 3c = 21 \\ b + c = 10 \end{cases}" />

Using elimination or substitution, we get <InlineMath math="b = 9" /> and <InlineMath math="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

<div className="flex flex-col gap-4">
  <LineEquation
    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"),
      },
    ]}
  />
  <LineEquation
    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"),
      },
    ]}
  />
  <LineEquation
    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]}
  />
</div>
