The speech and science clubs at a high school are planning fundraisers for a local nonprofit organization. This system of linear equations represents the total estimated amount each club will raise, $y$, after $x$ days of the fundraiser.
[tex]
\begin{array}{l}
y=120 x-150 \\
y=90 x
\end{array}
[/tex]
Open the graphing tool and use it to solve the system of equations. Type the correct answer in each box. Use numerals instead of words.
The solution to the system of equations is ( $\square$, $\square$ ).
Answer (2)
Set the two equations equal to each other: 120 x − 150 = 90 x .
Solve for x : 30 x = 150 , so x = 5 .
Substitute x = 5 into y = 90 x to find y = 450 .
The solution to the system of equations is ( 5 , 450 ) .
Explanation
Problem Analysis Let's analyze the problem. We have a system of two linear equations:
y = 120 x − 150 y = 90 x
We need to find the solution to this system, which represents the point (x, y) where the two lines intersect. This point will tell us after how many days ( x ) the total amount raised ( y ) by each club will be the same.
Setting the Equations Equal To solve the system, we can set the two equations equal to each other:
120 x − 150 = 90 x
Solving for x Now, let's solve for x :
120 x − 90 x = 150 30 x = 150 x = 30 150 x = 5
Solving for y Now that we have the value of x , we can substitute it into either equation to find the value of y . Let's use the second equation:
y = 90 x y = 90 ( 5 ) y = 450
The Solution So, the solution to the system of equations is ( 5 , 450 ) . This means that after 5 days, both clubs will have raised the same amount, which is $450.
Examples
Imagine you're planning a charity event with two different fundraising strategies. One strategy starts with a deficit but raises more money each day, while the other starts with no deficit but raises less each day. Solving a system of equations like this helps you determine when both strategies will yield the same amount of money, allowing you to compare their effectiveness over time. This type of problem is useful in business for comparing different investment options or project plans, helping you make informed decisions about when one option becomes more profitable than another.
The solution to the system of equations is (5, 450), indicating that after 5 days, both clubs will have raised 450. T hi s w a s f o u n d b yse tt in g t h ee q u a t i o n se q u a lt oe a c h o t h er an d so l v in g f or x , t h e n s u b s t i t u t in g ba c k t o f in d y.
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