Interpreting a Solution to a System of Linear Equations
Students graphed the growth rate over several weeks of two plants in their classroom. The equations of both plants are given where $x$ represents the time in weeks and $y$ represents the heights of the plants in inches.
Plant A: $y=1.8 x+3.1$
Plant B: $y=2.3 x+1.9$
Approximately how many weeks will it take for both plants to reach the same height? Round your answer to the nearest tenth.
A. 1.9 weeks
B. 2.4 weeks
C. 3.1 weeks
D. 7.4 weeks
Answer (2)
Set the equations for Plant A and Plant B equal to each other: 1.8 x + 3.1 = 2.3 x + 1.9 .
Subtract 1.8 x from both sides: 3.1 = 0.5 x + 1.9 .
Subtract 1.9 from both sides: 1.2 = 0.5 x .
Divide both sides by 0.5 to find x = 2.4 . The plants will be the same height after 2.4 weeks.
Explanation
Understanding the Problem We are given two equations representing the heights of two plants, Plant A and Plant B, over several weeks. We want to find the number of weeks it will take for both plants to reach the same height. This means we need to find the value of x for which the y values (heights) of both plants are equal.
Setting up the Equation The equation for Plant A is y = 1.8 x + 3.1 , and the equation for Plant B is y = 2.3 x + 1.9 . To find when the heights are equal, we set the two equations equal to each other: 1.8 x + 3.1 = 2.3 x + 1.9
Isolating x Now, we solve for x . First, subtract 1.8 x from both sides of the equation: 3.1 = 0.5 x + 1.9
Further Isolating x Next, subtract 1.9 from both sides: 1.2 = 0.5 x
Solving for x Finally, divide both sides by 0.5 :
x = 0.5 1.2 = 2.4
Final Answer The value of x is 2.4 . Since we are asked to round to the nearest tenth, and 2.4 is already to the nearest tenth, our answer is 2.4 weeks.
Examples
Imagine you're tracking the growth of two investments. Plant A represents a low-risk investment with steady but slow growth, while Plant B represents a higher-risk investment with faster initial growth. Finding the point where their values are equal helps you decide when it might be beneficial to switch from the low-risk to the high-risk investment, or vice versa, based on your financial goals. This type of problem is also applicable in business when comparing different marketing strategies or product sales growths to determine when one strategy overtakes another.
The two plants will reach the same height in approximately 2.4 weeks. Therefore, the answer is B. 2.4 weeks.
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