One starts out at 40 cm and will grow 10 cm every three months. The other sapling starts at 20 cm and will grow 10 cm every two months. After how many months will they be the same height.

Question

How tall will they be?

define variables

Write two equations

solve using substitution or elimination

Answer 1

The first thing we want to do is write two equations. Our variables will be height, h, and months, m. Where height is a function of months h(m). Also be sure to recall the equation for a linear function, y = mx + b.

One starts out at 40cm and will grow 10cm every three months.

The rate at which the plant grows is 10cm every 3 months. This is our slope (m), given by 10/3. The plant starts out at 40cm and is our y-intercept (b). So:

h(m) = (10/3)m + 40

The other starts at 20cm and will grow 10cm every two months.

By the same process as the previous question we get:

h(m) = (10/2)m + 20

Substitute to get m.

You could use elimination for this problem, but I prefer substitution. Notice both equations are equal to h(m). This means that, by the transitive property, these equations are equal to each other:

(10/3)m + 40 = (10/2)m + 20

Now we solve for m.

(10/3)m + 20 = (10/2)m Subtract 20 from both sides.

20 = (5/3)m Subtract (10/3) from both sides. Remember to get the lowest common denominator when subtracting fractions.

60 = 5m Multiply by three on both sides.

12 = m Divide by 5 on both sides.

Plug m back into equation to get h(m).

You can plug m back in to either of the original equations to get h(m). For this example, I'll use the second equation.

h(12) = (10/2)(12) + 20

h(12) = 60 + 20

h(12) = 80

This means that after 12 months they will be the same height, 80cm.

Answer 2

m represents the number of months

40 + 10(m/3) = 20 + 10(m/2)

Can you solve for m and answer?