Will They Have the Same Height? Yes Height: 12 centimeters
Practice makes perfect
From our experiment, we know that Plant A has a starting height of 8 centimeters and grows 1 centimeter each week.
Similarly, we know that Plant B has a starting height of 4 centimeters and grows 2 centimeters each week.
We will write a system of equations representing this situation. Let x represent the number of weeks that have passed and let y represent the height of the plant.
Number of Weeks Passed &- x Height of the Plant &- y
Now, Plant A is 8 centimeters tall and Plant B is 4 centimeters tall. Then Plant A will grow by 1 centimeter and Plant B will grow by 2 centimeters each week. Let's create a system of equations for this situation. The first equation will represent the height of Plant A and the second equation will represent the height of Plant B.
y= 1* x+ 8 & (I) y= 2* x+ 4 & (II)
Let's recall what we know about the number of solutions of a system of linear equations when we know the slopes and y-intercepts of the equations.
Solutions of a System of Linear Equations
Slopes
y-Intercepts
Number of Solutions
Same
Same
Infinitely many solutions
Same
Different
No solution
Different
Different or the same
One solution
We can show these relationships on a graph. We know that the solution to the system is the point of intersection of two lines.
Keeping all of this in mind, let's consider the exercise. We want to determine if the two plants will ever have the same height. Let's take a look at the system of equations we created in Part A.
y=x+8 & (I) y=2x+4 & (II)
We can identify the slopes and y-intercepts.
Equation
Slope-Intercept Form
Slope
y-intercept
y=x+8
y= 1x+ 8
1
(0, 8)
y=2x+4
y= 2x+ 4
2
(0, 4)
Based on what we just reviewed about the number of solutions of a system, we know that our system will have one solution. The plants will have the same height at some point in time. Finally, we will solve this system by graphing. To do so, we will graph both equations on one coordinate plane and look for a point of intersection.
The two lines intersect at (4,12). This means that in week 4, both plants will be 12 centimeters tall.
