grade 8 Linear Functions & Equations

Linear Functions & Equations

A linear function and a linear equation are the same relationship, just read two ways. Where the graph of y = ax + b meets the x-axis, y is 0, so that x is the root of ax + b = 0. The two-unknown equation ax + by + c = 0 is a line on the plane, and the point where two lines meet is the solution of the system. Drag slope a and intercept b and you will see the x-intercept sit at the equation's answer.

Intuition — Function ↔ Equation

Slope a2

y-intercept b-4

👀 Function vs Equation

①The graph of y = ax + b crosses the x-axis

②At that point y = 0 so ax + b = 0

③So the root of the linear equation = x-intercept of the line!

General Form ax + by + c = 0

b-1

Two-variable Equation

ax + by + c = 0 → a line in the plane

If b ≠ 0, solve for y to write it as a function

System Solution = Intersection

Core Relation

Solution (x, y) of the system = intersection of the two lines

The point where the two function graphs meet

💡 Three Cases

①Lines meet at one point → one solution

②Parallel lines (same slope, different intercept) → no solution

③Identical lines → infinitely many solutions

Lines of the Form x = k or y = k

📝 Special Lines

①x = 3 → vertical line parallel to the y-axis (NOT a function!)

②y = 2 → horizontal line (slope 0, still a function)

③x = k assigns infinitely many y for one x — fails the function definition

Work It Out

Example 1

Write the line 2x + y − 4 = 0 in the form y = ax + b.

Solve for y.

y = −2x + 4

Read the slope and y-intercept.

slope −2, y-intercept 4

▸ y = −2x + 4

A line ax + by + c = 0 becomes a linear function once solved for y.

Example 2

Find the intersection of the lines y = x + 1 and y = −x + 5.

At the intersection both y values are equal, so set them equal.

x + 1 = −x + 5 ⇒ 2x = 4

Substitute x = 2.

x = 2, y = 2 + 1 = 3

▸ (2, 3)

The intersection of two lines is the solution of their two equations.

Exam Wrap-up

Function ↔ Equation ↔ Graph

y = ax+b ↔ ax-y+b = 0 ↔ a line graph

Three views of the same relationship

Grade-8 school exam type

If the lines y = ax + 1 and y = 3x − 2 are parallel, what is the constant a?

①1

②2

③3

④4

⑤5

▸ ③ 3

Parallel lines have equal slopes.

parallel ⇔ slope a = 3

Compare the slopes.

a = 3

🎯 Exam Points

①Equation root = x-intercept (set y = 0)

②ax+by+c=0 in two variables → a line in the plane

③System solution = the intersection of two lines

④x = k is a vertical line — not a function

⑤Line positions: parallel if same slope and different intercept (a₁ = a₂, b₁ ≠ b₂)

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Linear Functions & Graphs

Triangle Properties

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