Linear Equations: System of Two Lines

Q: What is the slope-intercept form of a linear equation?

The slope-intercept form is $y = ax + b$, where **$a$ is the slope** and **$b$ is the y-intercept**.

Q: How do we express $y$ as a function of $x$?

When we rewrite a linear equation to isolate $y$ in the form $y = ax + b$, the value of $y$ is determined entirely by $x$. This allows us to use function notation and write $y = f(x)$.

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

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Slope-Intercept Form & Function Notation

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 the slopes (

a_1, a_2

) and the 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 directions 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

Learning Topics

Frequently Asked Questions

What is the slope-intercept form of a linear equation?

The slope-intercept form is

y = ax + b

, where $a$ is the slope and $b$ is the y-intercept.

How do we express

y

as a function of

x

When we rewrite a linear equation to isolate

y

in the form

y = ax + b

, the value of

y

is determined entirely by

x

. This allows us to use function notation and write

y = f(x)