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 (1)
Set the two equations equal to each other: 120 x − 150 = 90 x .
Solve for x : 30 x = 150 , so x = 5 .
Substitute the value of x into one of the equations to solve for y : y = 90 ( 5 ) = 450 .
The solution to the system of equations is ( 5 , 450 ) .
Explanation
Understanding the Problem We are given a system of two linear equations:
y = 120 x − 150 y = 90 x
where x represents the number of days of the fundraiser and y represents the total estimated amount each club will raise. We need to find the solution to this system of equations, which represents the point (x, y) where the two lines intersect.
Setting the Equations Equal To solve the system of equations, we can set the two equations equal to each other:
120 x − 150 = 90 x
Now, we solve for x .
Solving for x Subtract 90 x from both sides of the equation:
120 x − 90 x − 150 = 90 x − 90 x 30 x − 150 = 0
Add 150 to both sides:
30 x = 150
Divide both sides by 30:
x = 30 150 = 5
So, x = 5 .
Solving for y Now that we have the value of x , we can substitute it into either equation to solve for y . Let's use the second equation:
y = 90 x
Substitute x = 5 :
y = 90 ( 5 ) = 450
So, y = 450 .
The Solution The solution to the system of equations is ( x , y ) = ( 5 , 450 ) . This means that after 5 days of the fundraiser, both clubs will have raised a total of $450.
Examples
Systems of equations are useful in many real-world scenarios. For example, suppose you are planning a party and need to determine how many pizzas and drinks you can buy with a certain budget. If you know the cost of each pizza and drink, and you have a total budget, you can set up a system of equations to find the number of pizzas and drinks you can purchase. Similarly, in business, systems of equations can help determine the break-even point, where costs equal revenue.
