Linear Equations

Master linear equations by rewriting them in slope-intercept form, identifying the slope and y-intercept, and writing equations of lines.

Linear Equations

Learn to rewrite linear equations in slope-intercept form, identify the slope and y-intercept, and write equations of lines.

Practice Topics

1. Slope-Intercept Form

Rewrite linear equations in the form $y = ax + b$ , determine the slope $a$ , and find the y-intercept $b$ .

2. Slope of a Line

Understand slope as steps, calculate the slope between two points, and find the equation of a line.

3. System of Two Lines

Determine the number of solutions for a system of two linear equations by comparing their slopes ( $a_1, a_2$ ) and their y-intercepts ( $b_1, b_2$ ).

4. Analyzing Linear Functions

Analyze linear functions: rewrite the equation in slope-intercept form $y = ax + b$ , identify the slope, the y-intercept $(0, b)$ and x-intercept $\left(-\frac{b}{a}, 0\right)$ (when $a \neq 0$ ), determine whether the function is increasing, decreasing, or constant, and draw the graph.

5. Area of Triangle Formed by Three Lines

Calculate the area of the triangle formed by the pairwise intersections of three lines using geometric decomposition.

Learning Guide

1. Slope-Intercept Form

A linear equation can be written in multiple forms. One of the most commonly used forms is the slope-intercept form:

Linear Equation (Slope-Intercept Form)

y = ax + b

The slope-intercept form of a linear equation, where a represents the slope (also called the gradient) and b represents the y-intercept.

where:
- $a$ is the slope (also called the gradient), which describes the direction and steepness of the line.
- $b$ is the y-intercept, representing the y-coordinate of the point where the line crosses the y-axis, located at

(0, b)

.

Note: We will cover the slope in detail in the next topic.

Function Notation:
Once we isolate

y

in a linear equation (writing it as

y = ax + b

), we can express

y

in function notation as

y = f(x)

.

In our case, the linear equation

y = ax + b

becomes:

f(x) = ax + b

Using function notation is very useful. For example, it allows us to evaluate the function for specific values of

x

. To find the y-value when

x = 0

, we substitute

0

for

x

f(0) = a \cdot 0 + b = b

This shows that when

x = 0

, the function value is

b

, which corresponds directly to the y-intercept at the point

(0, b)

2. Slope of a Line

The slope (represented by $a$ in the equation

y = ax + b

) describes the steepness and direction of a line.

Slope Steps Visualizer

For every step right (+1), we go up by 1.5.

Understanding Slope as Steps

Think of the slope as taking steps on the graph:
- Every time we move $1$ unit to the right (increase

x

1

), we go up or down by $a$ units (change

y

a

).
- If

a

is positive (e.g.

a = 2

), we go up by $2$ for every step right.
- If

a

is negative (e.g.

a = -1

), we move down by 1 for every step right.

The Slope Formula & Delta ( $\Delta$ )

To find the slope between two points

(x_1, y_1)

and

(x_2, y_2)

, we use the formula:

Slope of a Line

a = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

Calculated from two points

(x_1, y_1)

and

(x_2, y_2)

on the line.

What is Delta ( $\Delta$ )?
The Greek letter $\Delta$ (Delta) represents the change or difference between two values:
-

\Delta y = y_2 - y_1

is the change in the vertical direction (often called the rise).
-

\Delta x = x_2 - x_1

is the change in the horizontal direction (often called the run).

This is why the slope is often described as "rise over run" (

\frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}

).

⚠️ Crucial Order Note:
It does not matter which point you choose as the first point

(x_1, y_1)

and which as the second point

(x_2, y_2)

—the calculated slope will be exactly the same. However, you must maintain the same subtraction order in both the numerator and the denominator! If you subtract the

y

-coordinates as

y_2 - y_1

, you must subtract the

x

-coordinates as

x_2 - x_1

. Mixing the order (like

\frac{y_2 - y_1}{x_1 - x_2}

) is incorrect and will result in the wrong sign.

Slope Between Two Points

Slope Calculation:

a = \frac{\Delta y}{\Delta x} = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2

Finding the y-intercept $b$ and the Equation of a Line

Once we have the slope

a

, we can find the y-intercept

b

and write the full equation using a point

(x_1, y_1)

on the line. Since the point lies on the line, its coordinates must satisfy the equation

y = ax + b

.

Example:
Suppose we know the slope is

a = 3

, and the line passes through the point

(2, 7)

Start with the equation form: $y = ax + b$
Since $a = 3$ , this gives: $y = 3x + b$
Substitute the point $(x_1, y_1) = (2, 7)$ (where $x_1 = 2$ and $y_1 = 7$ ):
$y_1 = 3x_1 + b \implies 7 = 3 \cdot 2 + b$
Solve for $b$ :
$7 = 6 + b \implies b = 1$
The full equation is: $\boxed{y = 3x + 1}$

Equation of a Line from Two Points

If we are given only two points, we can determine the entire equation by combining the two steps:
1. Calculate the slope

a

from the two points using the slope formula.
2. Use the calculated slope

a

and one of the points to find the y-intercept

b

(exactly as shown in the example above).

Parallel Lines

Two lines are parallel if they lie in the same plane and never intersect, no matter how far they are extended. They rise or fall by the same amount for every unit moved horizontally, which means they have the same slope (

a_1 = a_2

), but different y-intercepts (

b_1 \neq b_2

).

Example:
- Line 1:

y = 2x + 3

(slope

a = 2

, y-intercept

b = 3

)
- Line 2:

y = 2x - 5

(slope

a = 2

, y-intercept

b = -5

)
Both lines have a slope of

2

, so they are parallel. Because they cross the y-axis at different heights (

3

and

-5

), they are distinct, non-overlapping lines that will never touch.

Parallel Lines Visualizer

Select Shared Slope (a):

Blue Intercept (b₁):

Red Intercept (b₂):

y = 1.5x + 1

y = 1.5x - 2

Property of Parallel Lines

The graphs of two linear functions are parallel if and only if they have the same slope:

a_1 = a_2

(and different y-intercepts:

b_1 \neq b_2

Note: "If and only if" means the rule works in both directions: if the lines are parallel, their slopes must be equal; and if their slopes are equal, the lines must be parallel.

3. System of Two Lines

In addition to solving a system of two linear equations (

y = a_1 x + b_1

and

y = a_2 x + b_2

) algebraically (see the Classifying Systems of Equations topic), we can determine their geometric relationship and the number of solutions directly by comparing their slopes (

a_1, a_2

) and their y-intercepts (

b_1, b_2

Relative Positions Visualizer

(1) y = 1.5x + 1

(2) y = -0.5x + 2

1. Intersecting Lines (Exactly One Solution)

If the slopes are different (

a_1 \neq a_2

), the lines have different slopes and intersect at exactly one point. The system has one unique solution.

2. Parallel Lines (No Solution)

If the slopes are equal (

a_1 = a_2

), but the y-intercepts are different (

b_1 \neq b_2

), the lines run parallel to each other at a constant distance and never meet. The system has no solution.

3. Coincident Lines (Infinitely Many Solutions)

If both the slopes and the y-intercepts are identical (

a_1 = a_2

and

b_1 = b_2

), the two equations represent the exact same line. Every point on the line is a shared point, so the system has infinitely many solutions.

Mastering SealMath: Typing Subscripts ( $a_1$ and $a_2$ )

To enter subscript parameters like

a_1

a_2

(or

b_1

and

b_2

) in the math input, type a_1 or a_2 using the underscore key (_) on your keyboard, or click the subscript button (

x_n

) on the virtual math keyboard. Note: Typing a1 or a2 without an underscore is not accepted, because in mathematics

a1

represents

a \cdot 1

and

a2

represents

2a

4. Analyzing Linear Functions

Analyzing a linear function involves converting its equation into slope-intercept form

y = ax + b

, determining its main geometric properties (slope, x- and y-intercepts, increasing, decreasing, or constant behavior), and drawing its graph.

Linear Function Analysis Visualizer

Slope (a):

Y-intercept (b):

Equation:

y = 2x - 2

Slope (a): 2

Y-intercept: (0, -2)

X-intercept: (1, 0)

Behavior: Increasing

Crossed Quadrants: 1, 3, 4

Plotted Points: (4)

Drawing Checklist:

✓ Y-intercept (0, b)
✓ X-intercept $\left(-\frac{b}{a}, 0\right)$ (if $a \neq 0$)
✓ Point in Quadrant 1
✓ Point in Quadrant 3
○ Point in Quadrant 4

1. Derive the Slope-Intercept Form ( $y = ax + b$ )

Start by isolating

y

on the left side of the given linear equation.
Example: Given

2x + 3y - 6 = 0

3y = -2x + 6 \implies y = -\frac{2}{3}x + 2

2. Identify the Slope ( $a$ )

The coefficient

a

y = ax + b

is the slope (gradient) of the line, which describes the line's steepness and direction.
In our example

y = -\frac{2}{3}x + 2

, the slope is

a = -\frac{2}{3}

3. Find Intercepts with the Coordinate Axes

- Y-intercept (intersection with the y-axis): Substitute

x = 0

into the equation:

y = a(0) + b = b \implies \text{Point } (0, b)

- X-intercept (intersection with the x-axis): Substitute

y = 0

into the equation:

0 = ax + b \implies ax = -b \implies x = -\frac{b}{a} \implies \text{Point } \left(-\frac{b}{a}, 0\right)

(If $b = 0$ , both intercepts coincide at the origin $(0, 0)$ . If $a = 0$ and $b \neq 0$ , the line $y = b$ is horizontal and has no x-intercept).

4. Determine Function Behavior (Increasing / Decreasing / Constant)

- Increasing: If slope

a > 0

y

increases as

x

increases.
- Decreasing: If slope

a < 0

y

decreases as

x

increases.
- Constant: If slope

a = 0

y

remains constant for all

x

💡 Note on Degree-0 Constant Functions ( $y = c$ )

A constant function

y = c

(where slope

a = 0

) is a special case of a linear function with a horizontal graph. In polynomial terminology, it is a degree-0 polynomial.

5. Drawing the Function

To draw the function graph on the coordinate plane:
1. Plot the x- and y-intercepts: the y-intercept

(0, b)

and the x-intercept

\left(-\frac{b}{a}, 0\right)

.
2. Plot at least one additional point in each quadrant that the line passes through.

❓ Frequently Asked Questions

How do you analyze a linear function?

To analyze a linear function

y = ax + b

, follow these main steps:
1. Slope-Intercept Form: Isolate

y

to rewrite the equation as

y = ax + b

.
2. Parameters: Identify the slope

a

and the Y-intercept value

b

.
3. X- and Y-Intercepts: Calculate the y-intercept at

(0, b)

and the x-intercept at

\left(-\frac{b}{a}, 0\right)

(when

a \neq 0

).
4. Function Behavior: Determine whether the function is increasing (

a > 0

), decreasing (

a < 0

), or constant (

a = 0

).
5. Graphing: A line is completely determined by two distinct points. Since the exercise checks the required points rather than a continuous drawn line, plot the x- and y-intercepts and the required additional points that lie on the line.

5. Area of Triangle Formed by Three Lines

When three distinct lines intersect pairwise at three distinct points, they form a triangle. To determine its area, we find the three intersection vertices and apply geometric decomposition with a reference right triangle.

Triangle Area Decomposition Visualizer

Lines:

L_1: y = 0, \quad L_2: y = -2x + 8, \quad L_3: y = 2x

Triangle vertices:

A(0,0), B(4,0), C(2,4)

Additive Area Calculation:

S_{\Delta ABC} = S_{ADC} + S_{BDC} = \frac{2 \cdot 4}{2} + \frac{2 \cdot 4}{2} = 4 + 4 = 8

1. Calculate the Three Vertices

Solve pairwise linear systems to determine the three intersection coordinates:

A(x_A, y_A)

B(x_B, y_B)

, and

C(x_C, y_C)

2. Construct a Reference Right-Angled Triangle

Draw a horizontal or vertical auxiliary line through one vertex to construct a right-angled reference triangle alongside two additional triangles.

Additive Method

Given vertices $A(0,0)$ , $B(4,0)$ , $C(2,4)$ :

Auxiliary Line: An auxiliary vertical line through $C(2,4)$ intersects base $AB$ at $D(2,0)$ .
Sub-regions: This divides the triangle into two adjacent right triangles:
- $S_1 = \triangle ADC$ : base $AD = 2$ , height $CD = 4 \implies S_1 = \frac{2 \cdot 4}{2} = 4$
- $S_2 = \triangle BDC$ : base $BD = 2$ , height $CD = 4 \implies S_2 = \frac{2 \cdot 4}{2} = 4$
Calculation: $S_{\Delta ABC} = S_1 + S_2 = S_{ADC} + S_{BDC} = 4 + 4 = 8$

Subtractive Method 1: Reference Triangle Subtraction

Given vertices $A(0,0)$ , $B(4,4)$ , $C(3,1)$ :

Reference Right Triangle: Construct auxiliary point $D(4,0)$ on the x-axis to form reference right-angled triangle $\triangle ABD$ with base $AD = 4$ , height $BD = 4$ , and area $S_{\text{ref}} = \frac{4 \cdot 4}{2} = 8$ .
Triangles to Subtract:
- $S_1 = \triangle BCD$ : base $BD = 4$ , height $= 1 \implies S_1 = \frac{4 \cdot 1}{2} = 2$
- $S_2 = \triangle ACD$ : base $AD = 4$ , height $= 1 \implies S_2 = \frac{4 \cdot 1}{2} = 2$
Calculation: $S_{\Delta ABC} = S_{\text{ref}} - S_1 - S_2 = S_{\Delta ABD} - S_{\Delta BCD} - S_{\Delta ACD} = 8 - 2 - 2 = 4$

Subtractive Method 2: Bounding Box Subtraction

Given vertices $A(1,1)$ , $B(5,2)$ , $C(2,5)$ :

Bounding Box: Enclose the triangle inside a bounding box from $(1,1)$ to $(5,5)$ with width $4$ , height $4$ , and area $S_{\text{box}} = 4 \cdot 4 = 16$ .
Exterior Triangles to Subtract:
- $S_1$ : base $= 4$ , height $= 1 \implies S_1 = \frac{4 \cdot 1}{2} = 2$
- $S_2$ : base $= 3$ , height $= 3 \implies S_2 = \frac{3 \cdot 3}{2} = 4.5$
- $S_3$ : base $= 1$ , height $= 4 \implies S_3 = \frac{1 \cdot 4}{2} = 2$
Calculation: $S_{\Delta ABC} = S_{\text{box}} - S_1 - S_2 - S_3 = 16 - 2 - 4.5 - 2 = 7.5$

❓ Frequently Asked Questions

How do you find the area of a triangle formed by three lines?

1. Find the 3 intersection points by solving pairwise linear equations.
2. Draw a horizontal or vertical auxiliary line through a vertex to construct a reference right-angled triangle.
3. Compute the area using Additive Method (

S_1 + S_2

) or Subtractive Method (

S_{\text{ref}} - S_1 - S_2

) depending on the geometry.

Learning Topics