Intersection and Zero of Linear Functions
The zero of a linear function is the x-value where the line crosses the x-axis, and the intersection of two linear functions is the point where their lines meet. In brief: for f(x) = 2x − 8 the zero is x = 4, since 2x − 8 = 0. The lines y = x + 1 and y = −x + 5 intersect at (2, 3), found by solving x + 1 = −x + 5.
Both ideas reduce to solving a linear equation. They answer practical questions: when does a balance reach zero, when do two prices become equal, when do two travellers meet? This guide shows how to find zeros and intersections algebraically and graphically, how to handle the cases of no solution and infinitely many solutions and how to use the results in problems.
How do you find the zero of a linear function?
The zero of f(x) = mx + b is the solution of mx + b = 0. For m ≠ 0 it is x = −b/m, and the x-intercept is (−b/m, 0). If m = 0 and b ≠ 0, the function is a nonzero constant and has no zero. If m = 0 and b = 0, every x is a zero.
Example 1
f(x) = −3x + 12. Solve −3x + 12 = 0, so x = 4. The line crosses the x-axis at (4, 0), and the y-intercept is (0, 12).
Example 2: fraction
f(x) = (2/5)x − 3. Then (2/5)x = 3, so x = 15/2 = 7.5.
The zero is also where the sign of the function changes. For f(x) = 2x − 8, the values are negative for x < 4 and positive for x > 4. This is used when solving linear inequalities.
How do you find the intersection of two linear functions?
- Set the two expressions equal: m1x + b1 = m2x + b2.
- Solve for x: x = (b2 − b1)/(m1 − m2).
- Substitute x into either equation to find y.
- Check by using the other equation.
Example 3
f(x) = 3x − 1 and g(x) = −x + 7. Setting 3x − 1 = −x + 7 gives 4x = 8, so x = 2. Then f(2) = 5, and g(2) = 5 ✓. The intersection is (2, 5).
Example 4: from standard form
For 2x + y = 8 and x − y = 1, solve the first for y: y = 8 − 2x. The second gives y = x − 1. Setting 8 − 2x = x − 1 gives 9 = 3x, so x = 3 and y = 2. This is the substitution method.
How many intersection points can two lines have?
- One point: the slopes are different.
- No point: the slopes are equal and the intercepts differ (parallel lines). The equation reduces to a false statement such as 0 = 4.
- Infinitely many points: the slopes and intercepts are both equal (the same line). The equation reduces to a true statement such as 0 = 0.
For y = 2x + 3 and y = 2x − 1: 2x + 3 = 2x − 1 gives 3 = −1, which is false, so there is no intersection. For y = 2x + 3 and 2y = 4x + 6, the second line is the same as the first.
How do you use intersections in word problems?
Define a function for each situation with the same input variable, set them equal and solve. Interpret the answer in context and give the units.
Example 5: comparing plans
Gym A charges a 30 dollar joining fee and 20 dollars per month: A(t) = 20t + 30. Gym B has no fee and charges 35 dollars per month: B(t) = 35t. Setting 20t + 30 = 35t gives 15t = 30, so t = 2 months. After 2 months both cost 70 dollars. For longer periods, Gym A is cheaper: at t = 6, A = 150 and B = 210.
Example 6: meeting travellers
Anna starts at 0 km walking at 5 km/h: s = 5t. Ben starts at 12 km from her walking towards her at 4 km/h: s = 12 − 4t. They meet when 5t = 12 − 4t, so 9t = 12 and t = 4/3 h, which is 1 hour 20 minutes, at s = 20/3 ≈ 6.67 km.
Examples of zeros and intersections
| Function(s) | Equation to solve | Result |
|---|---|---|
| f(x) = 2x − 8 | 2x − 8 = 0 | zero x = 4 |
| f(x) = −x + 6 | −x + 6 = 0 | zero x = 6 |
| y = x + 1 and y = −x + 5 | x + 1 = −x + 5 | (2, 3) |
| y = 3x and y = 2x + 4 | 3x = 2x + 4 | (4, 12) |
| y = 2x + 3 and y = 2x − 1 | 3 = −1 | no intersection |
| y = 2x + 3 and 2y = 4x + 6 | 0 = 0 | same line |
Where do these ideas come from?
Problems about meeting points and equal values are very old. Ancient problems of the type “when will two things be equal” were solved by the rule of false position, which is a method for linear equations, in Egyptian and later in Indian and Arabic mathematics. Coordinate geometry later turned them into the intersection of lines.
Where are they used today?
Break-even analysis finds the intersection of cost and revenue lines. Scheduling asks when two moving objects are at the same place. Price comparisons, such as mobile tariffs, are intersection problems. In computing, determining whether two line segments cross uses this algebra, for example in collision detection.
Which mistakes are common?
Giving only the x-coordinate
An intersection is a point. Compute y as well.
Setting the wrong things equal
For an intersection, set f(x) = g(x). For a zero, set f(x) = 0.
Sign errors when moving terms
From 3x − 1 = −x + 7, adding x to both sides gives 4x − 1 = 7. Do not drop the sign.
Interpreting “no solution” as a mistake
Parallel lines have no intersection. A false statement like 3 = −1 is a valid answer to the question.