A linear function and a linear equation are one relationship seen through different lenses.
Where y = ax + b crosses the x-axis, y is 0, so that x-value is the root of ax + b = 0.
The two-unknown equation ax + by + c = 0 draws a line, and two lines meeting gives a system's solution.
Here you drag the slope and intercept to watch the x-intercept become the equation's answer.
Intuition — Function ↔ Equation
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Graph showing the x-intercept of a linear function as the equation root
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👀 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
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See how ax + by + c = 0 changes shape as coefficients change
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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.
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Solve for y.
y = −2x + 4
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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.
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At the intersection both y values are equal, so set them equal.
x + 1 = −x + 5 ⇒ 2x = 4
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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?