When Will the Trees Be the Same Height? Linear Equations Problem

Linear Equations 9th-10th Grade
PROBLEM
When Michael moved into a new house, he planted two trees in his backyard. At the time of planting, Tree A was 21 inches tall and Tree B was 33 inches tall. Each year thereafter, Tree A grew by 8 inches per year and Tree B grew by 5 inches per year. Write an equation for each situation, in terms of t, and determine the height of both trees at the time when they have an equal height.

What This Problem Teaches

  • Writing linear equations from real-world growth scenarios
  • Setting up equations where one variable appears in multiple expressions
  • Understanding "catch-up" problems where the faster rate starts behind
  • Solving systems by substitution when equations are already solved for the same variable
  • Interpreting intersection points as moments of equality in time-based problems

Visualizing the Growth

When Michael moved into a new house, he planted two trees in his backyard. At the time of planting, Tree A was 21...

The graph shows both trees' growth over time. Tree A starts shorter but grows faster, eventually catching up to Tree B after 4 years when both reach 53 inches.

Solution: The Equation Approach

Step 1 — Write the growth equation for each tree

Each tree's height follows the pattern: Current Height = Initial Height + Growth Rate × Time

Tree A: Starts at 21 inches, grows 8 inches per year

A(t) = 21 + 8t

Tree B: Starts at 33 inches, grows 5 inches per year

B(t) = 33 + 5t

Step 2 — Set the heights equal to find when they're the same

When the trees have equal height, A(t) = B(t):

21 + 8t = 33 + 5t

Step 3 — Solve for t

Subtract 5t from both sides:

21 + 3t = 33

Subtract 21 from both sides:

3t = 12

Divide by 3:

t = 4

Step 4 — Find the height at that time

Substitute t = 4 into either equation. Using Tree A's equation:

A(4) = 21 + 8(4) = 21 + 32 = 53 inches

Solution: Method 2 — The Gap-Closing Analysis

Step 1 — Find the initial gap between trees

Tree B starts ahead by: 33 - 21 = 12 inches

Step 2 — Calculate how fast Tree A closes the gap

Each year, Tree A gains: 8 - 5 = 3 inches on Tree B

Step 3 — Find when the gap closes completely

Time to close 12-inch gap at 3 inches per year:

t = 12 ÷ 3 = 4 years

Step 4 — Calculate the height at that time

After 4 years, Tree A's height:

21 + 8(4) = 53 inches
The trees will have equal height after 4 years, when both trees are 53 inches tall.

Tree A equation: A(t) = 21 + 8t
Tree B equation: B(t) = 33 + 5t

Verification

Let's check both trees at t = 4 years:

Tree A:A(4) = 21 + 8(4) = 21 + 32 = 53 inches ✓

Tree B:B(4) = 33 + 5(4) = 33 + 20 = 53 inches ✓

Both trees are exactly 53 inches tall after 4 years, confirming our answer.

We can also verify the pattern makes sense: Tree A starts 12 inches behind but gains 3 inches per year on Tree B. After 4 years, Tree A gains 4 × 3 = 12 inches on Tree B, exactly closing the initial gap.

Common Pitfalls

✗ Mistake: Writing the equations as A(t) = 8t and B(t) = 5t

Why it's wrong: This ignores the initial heights and assumes both trees start at 0 inches. The trees don't start from nothing—they begin with heights of 21 and 33 inches respectively.

✗ Mistake: Solving 21 + 8t = 33 + 5t and getting t = -4

Why it's wrong: This happens when you incorrectly move terms: 8t - 5t = 33 - 21 gives 3t = 12, not -3t = 12. Negative time doesn't make sense in this context—we're looking forward from planting, not backward.

✗ Mistake: Adding the growth rates: "They grow 8 + 5 = 13 inches per year combined"

Why it's wrong: We're not looking for combined growth—we want to know when individual heights are equal. The trees grow independently, and we need to track each one separately until they meet.

The Pattern Behind This

This is a classic "catch-up" linear equation problem. The general pattern is:

Object 1: h₁(t) = initial₁ + rate₁ × t
Object 2: h₂(t) = initial₂ + rate₂ × t
Equal when: initial₁ + rate₁ × t = initial₂ + rate₂ × t

The solution follows a predictable structure:

  • Gap: |initial₂ - initial₁| = how far apart they start
  • Closing rate: |rate₁ - rate₂| = how fast the gap changes
  • Catch-up time: Gap ÷ Closing rate = when they meet

Important note: This only works when the faster rate belongs to the object starting behind. If the object starting ahead also grows faster, they'll never meet!

Real Applications

This catch-up pattern appears frequently in real scenarios:

  • Population growth: Smaller cities growing faster than larger ones, eventually reaching equal populations
  • Investment accounts: An account starting with less money but earning higher interest, catching up to a larger account with lower returns
  • Manufacturing: Two production lines with different starting inventories but different production rates reaching equal output

Four "What-If?" Problems

1

Different Starting Gap

Tree C starts at 15 inches and grows 9 inches per year. Tree D starts at 40 inches and grows 4 inches per year. When will they be the same height, and what will that height be?
Step 1 — Write the equations

Tree C: C(t) = 15 + 9t
Tree D: D(t) = 40 + 4t

Step 2 — Set equal and solve

15 + 9t = 40 + 4t
5t = 25
t = 5

Step 3 — Find the height

C(5) = 15 + 9(5) = 60 inches

Verification

D(5) = 40 + 4(5) = 60 inches ✓

Answer: 5 years, 60 inches tall

2

Reverse Engineering

Two trees start at 25 inches and 45 inches. After 8 years, they are the same height. If the first tree grows 3 inches per year, what must be the growth rate of the second tree?
Step 1 — Set up known information

Tree 1: 25 + 3(8) = 25 + 24 = 49 inches after 8 years
Tree 2: 45 + r(8) where r is unknown growth rate

Step 2 — Set heights equal at t = 8

49 = 45 + 8r

Step 3 — Solve for r

4 = 8r
r = 0.5 inches per year

Verification

Tree 2 after 8 years: 45 + 0.5(8) = 49 inches ✓

Answer: 0.5 inches per year

3

Three Trees Competition

Use the original trees (A: 21 inches, 8 in/yr; B: 33 inches, 5 in/yr) plus Tree C planted at the same time: 10 inches tall, growing 11 inches per year. When do Trees A and C reach the same height?
Step 1 — Write equations for A and C

Tree A: A(t) = 21 + 8t
Tree C: C(t) = 10 + 11t

Step 2 — Set equal

21 + 8t = 10 + 11t

Step 3 — Solve

21 - 10 = 11t - 8t
11 = 3t
t = 11/3 ≈ 3.67 years

Step 4 — Find height

A(11/3) = 21 + 8(11/3) = 21 + 88/3 = 151/3 ≈ 50.33 inches

Verification

C(11/3) = 10 + 11(11/3) = 10 + 121/3 = 151/3 inches ✓

Answer: 3⅔ years, 50⅓ inches tall

4

Growth Rate Changes

Tree A starts at 21 inches and grows 8 inches per year for the first 3 years, then slows to 4 inches per year. Tree B grows consistently at 5 inches per year from its 33-inch start. Do they ever become equal height? If so, when?
Step 1 — Find Tree A's height after 3 years

A(3) = 21 + 8(3) = 45 inches

Step 2 — Find Tree B's height after 3 years

B(3) = 33 + 5(3) = 48 inches

Step 3 — Set up equations for t > 3

Let s = years after year 3
Tree A: 45 + 4s
Tree B: 48 + 5s

Step 4 — Check if they can be equal

45 + 4s = 48 + 5s
45 - 48 = 5s - 4s
-3 = s

Step 5 — Interpret the result

Since s = -3 is negative, this equality would have occurred 3 years before year 3, which contradicts our setup. After Tree A slows down, Tree B pulls further ahead each year.

Answer: No, they never become equal height after the growth rate change.

Frequently Asked Questions

How do you write a linear equation for a growth problem?+
Use the form y = initial value + (rate × time). The initial value is the starting height, the rate is the growth per year, and time is the variable. In this example, Tree A starts at 21 inches and grows 8 inches per year, so its equation is A(t) = 21 + 8t.
When will two objects growing at different rates be the same size?+
Set their equations equal to each other and solve for time. The faster-growing object will eventually catch up to the slower one if it started smaller. In this problem, 21 + 8t = 33 + 5t leads to t = 4 years.
Why does the faster rate sometimes start behind in these problems?+
This creates an interesting "catch-up" scenario where the underdog eventually overtakes the leader. It models real situations like younger siblings eventually becoming taller, or small companies growing faster than established ones. Here, Tree A starts 12 inches shorter but grows 3 inches per year faster.
NJ
Neven Jurkovic, PhD

Professor of Computer Science, Palo Alto College, Alamo Colleges District, San Antonio, TX

Developer of Algebrator

Contact

This solution was prepared with AI assistance and reviewed by Dr. Jurkovic for mathematical accuracy and pedagogical clarity.

2026-07-14

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