A linear function y = ax + b and a linear equation ax + b = 0 are the same expression read in two directions.
Where the graph meets the x-axis, y is 0, so the x-coordinate of that point is exactly the solution of the equation.
An equation in two unknowns, ax + by + c = 0, also draws a straight line, and the point where two such lines cross gives the solution of the system.
In the first picture, changing the slope or the y-intercept moves the red dot on the x-axis, and that dot is the answer to the equation.
The x-intercept is the solution
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A line that moves with its slope and y-intercept, and the point where it crosses the x-axis
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👀 Solving an equation with a graph
①Where the line y = ax + b meets the x-axis, y = 0.
②So the x-coordinate of that point satisfies ax + b = 0: the solution of the equation is the x-intercept of the graph.
③If the slope is 0, the line runs parallel to the x-axis. With b ≠ 0 it never meets the axis and there is no solution; with b = 0 it lies on the axis and every x is a solution.
Graphs of equations in two unknowns
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The line ax + by + c = 0 and its two intercepts as a, b and c change
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Linear equation in two unknowns
ax + by + c = 0 → a straight line in the coordinate plane
If b ≠ 0, solving for y gives y = −(a/b)x − c/b, a linear function. If b = 0, it becomes the vertical line x = −c/a.
Solutions of a system and intersection points
Key relation
solution (x, y) of the system = intersection of the two lines
A point that satisfies both equations lies on both lines at once.
💡 Three possible cases
①Different slopes: the lines meet at exactly one point, so there is one solution.
②Same slope, different y-intercepts: the lines are parallel and never meet, so there is no solution.
③Same slope and same y-intercept: the lines coincide, so there are infinitely many solutions.
Lines of the form x = k and y = k
📝 Horizontal and vertical lines
①y = 2 is the same as 0·x + y − 2 = 0: a horizontal line parallel to the x-axis, and a function with slope 0.
②x = 3 is the same as x + 0·y − 3 = 0: a vertical line parallel to the y-axis.
③On x = 3 a single x is paired with infinitely many y values, so it is not a function. Setting b to 0 in the picture above produces exactly such a vertical line.
Worked examples
Example 1
Write the line 2x + y − 4 = 0 in the form y = ax + b.
1
Move 2x and −4 to the right-hand side so that only y is left.
y = −2x + 4
2
Read the slope and y-intercept.
slope −2, y-intercept 4
▸ y = −2x + 4
Solving ax + by + c = 0 for y puts the slope and the y-intercept in plain view.
Example 2
Find the intersection of the lines y = x + 1 and y = −x + 5.
1
At the intersection both lines have the same y, so set the right-hand sides equal.
x + 1 = −x + 5 ⇒ 2x = 4 ⇒ x = 2
2
Substitute x = 2 into one of the equations to find y.
y = 2 + 1 = 3
▸ (2, 3)
The intersection of two lines is the solution of the system formed by their equations.
Exam checklist
Three ways to write one relation
y = ax + b ↔ ax − y + b = 0 ↔ a line in the plane
Read the same relation as a function, as an equation and as a graph.
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
1
Parallel lines have equal slopes and different y-intercepts.
a = 3
2
The y-intercepts 1 and −2 differ, so the lines do not coincide.
1 ≠ −2 ⇒ parallel
🎯 Exam points
①The solution of ax + b = 0 is the x-intercept of y = ax + b (substitute y = 0).
②The graph of ax + by + c = 0 is a straight line.
③The solution of a system is the intersection point of the two lines.
④x = k is a vertical line and not a function; y = k is a function with slope 0.
⑤y = a₁x + b₁ and y = a₂x + b₂ are parallel when a₁ = a₂ and b₁ ≠ b₂.