Question

a teacher purchased 20 calculators and 10 measuring tapes for her class and paid $495. later, she realized that she didnt order enough supplies. she placed another order of 8 of the same calculators and 1 more of the same measuring tape and paid $178.50. this system represents the constraints in this situation: {20c + 10m = 495, 8c + m = 178.50} 1 discuss with a partner: a. in this situation, what do the solutions to the first equation mean? b. what do the solutions to the second equation mean? c. for each equation, how many possible solutions are there? explain how you know. d. in this situation, what does the solution to the system mean? 2 find the solution to the system. explain or show your reasoning.

Explanation:

Step1: Interpret first - equation meaning

The equation \(20c + 10m=495\) represents the total cost of the first - order. Here, \(c\) is the cost of one calculator and \(m\) is the cost of one measuring tape. The solutions \((c,m)\) to this equation are all pairs of calculator and measuring - tape costs such that the cost of 20 calculators and 10 measuring tapes is $495.

Step2: Interpret second - equation meaning

The equation \(8c + m=178.50\) represents the total cost of the second - order. The solutions \((c,m)\) to this equation are all pairs of calculator and measuring - tape costs such that the cost of 8 calculators and 1 measuring tape is $178.50.

Step3: Determine number of solutions for each equation

Each of the given equations is a linear equation in two variables (\(c\) and \(m\)). A linear equation in two variables \(Ax+By = C\) (where \(A\), \(B\), and \(C\) are constants and \(A\) and \(B\) are not both zero) has infinitely many solutions. This is because for any value of \(x\) (in our case \(c\)), we can find a corresponding value of \(y\) (in our case \(m\)) that satisfies the equation.

Step4: Interpret system solution meaning

The solution to the system of equations \(

$$\begin{cases}20c + 10m=495\\8c + m=178.50\end{cases}$$

\) represents the cost of one calculator (\(c\)) and the cost of one measuring tape (\(m\)) that satisfy both the cost of the first order and the cost of the second order simultaneously.

Step5: Solve the system

From the second equation \(m = 178.50-8c\).
Substitute \(m\) into the first equation:
\[

$$\begin{align*} 20c+10(178.50 - 8c)&=495\\ 20c+1785-80c&=495\\ - 60c&=495 - 1785\\ -60c&=-1290\\ c&=\frac{-1290}{-60}=21.5 \end{align*}$$

\]
Substitute \(c = 21.5\) into \(m = 178.50-8c\):
\[

$$\begin{align*} m&=178.50-8\times21.5\\ &=178.50 - 172\\ &=6.5 \end{align*}$$

\]

Answer:

a. The solutions \((c,m)\) to \(20c + 10m = 495\) are all pairs of calculator and measuring - tape costs such that the cost of 20 calculators and 10 measuring tapes is $495.
b. The solutions \((c,m)\) to \(8c + m=178.50\) are all pairs of calculator and measuring - tape costs such that the cost of 8 calculators and 1 measuring tape is $178.50.
c. Each equation has infinitely many solutions because they are linear equations in two variables. For any value of one variable, we can find a corresponding value of the other variable that satisfies the equation.
d. The solution \((c,m)=(21.5,6.5)\) means the cost of one calculator is $21.5 and the cost of one measuring tape is $6.5.

1. The solution to the system is \(c = 21.5\) and \(m = 6.5\). The reasoning is shown above by substitution method to solve the system of linear equations.