Systems of linear equations and inequalities

2000004004

Level: 
A
Identify which of the following systems of equations has no solution.
\[\begin{aligned} y & = 10-2x & & \\ y & = 5 -2x& & \end{aligned}\]
\[\begin{aligned} y & = 10-2x & & \\ 2y & = 20 -4x& & \end{aligned}\]
\[\begin{aligned} y & = 10-2x & & \\ -y & = 5 -2x& & \end{aligned}\]
\[\begin{aligned} y & = 10-2x & & \\ 3y & = 30 -6x& & \end{aligned}\]

2000004003

Level: 
A
Identify which of the following systems of equations has no solution.
\[\begin{aligned} x + 3y & = 11 & & \\5x +15y & = 33 & & \end{aligned}\]
\[\begin{aligned} x + 3y & = 11 & & \\5x +15y & = 55 & & \end{aligned}\]
\[\begin{aligned} x + 3y & = 11 & & \\3x +12y & = 33 & & \end{aligned}\]
\[\begin{aligned} x + 3y & = 11 & & \\-x +3y & = 11 & & \end{aligned}\]

2000004002

Level: 
A
Identify which of the following systems of equations has infinitely many solutions.
\[\begin{aligned} 2y & = 5x-3 & & \\ y & = \frac{5}{2}x-\frac{3}{2} & & \end{aligned}\]
\[\begin{aligned} 2y & = 5x-3 & & \\ y & = \frac{5}{3}x-\frac{3}{2} & & \end{aligned}\]
\[\begin{aligned} 2y & = 5x-3 & & \\ y & = \frac{5}{4}x-1 & & \end{aligned}\]
\[\begin{aligned} 2y & = 5x-3 & & \\ -y & = \frac{5}{2}x+\frac{3}{2} & & \end{aligned}\]

2000004001

Level: 
A
Identify which of the following systems of equations has infinitely many solutions.
\[\begin{aligned} x - y & = 5 & & \\2x - 2y & = 10 & & \end{aligned}\]
\[\begin{aligned} x - y & = 5 & & \\3x - 3y & = 10 & & \end{aligned}\]
\[\begin{aligned} x - y & = 5 & & \\-x +y & = 5 & & \end{aligned}\]
\[\begin{aligned} x - y & = 5 & & \\2x +2y & = 10 & & \end{aligned}\]

1103034507

Level: 
B
Consider a balance scales comprising of beam with unequal length of arms where the fulcrum is very close to one end of the beam. (Such scales are called steelyard. For instance, it is often used to weigh a catch in fisheries.) The load is hanged on the shorter arm, while the balance about the fulcrum is obtained by sliding the counterweight along the longer arm. (See the picture.) Suppose the distance of the load hanging point from the fulcrum is fixed at \( 5\,\mathrm{cm} \). If the weight of the load is \( 80\,\mathrm{N} \), the balance is achieved as the counterweight is moved to the very end of the longer arm. If the weight of the load is \( 60\,\mathrm{N} \), the balance is achieved when the counterweight is moved to the distance of \( 30\,\mathrm{cm} \) from the fulcrum. What is the length of the beam? \[ \] Hint: The steelyard is based on the law of the lever. For balanced lever is: \( F_1\cdot a=F_2\cdot b \), where \( F_1 \) is the weight of the load in the distance \( a \) from the fulcrum and \( F_2 \) is the weight of the counterweight in the distance \( b \) from the fulcrum.
\( 45\,\mathrm{cm} \)
\( 54\,\mathrm{cm} \)
\( 40\,\mathrm{cm} \)
\( 35\,\mathrm{cm} \)

1003034506

Level: 
B
Kamil is able to mow a meadow in \( 12 \) hours. Zdeněk has a better lawn mower and he is able to mow the same meadow in \( 8 \) hours. They have agreed that Kamil starts to mow alone sooner and Zdeněk will join him later so that the total time of mowing is \( 9 \) hours. How long will they mow together?
\( 2 \) hours
\( 7 \) hours
\( 6 \) hours
\( 3 \) hours

1003034505

Level: 
B
The March price of a T-shirt and shorts was \( 600\,\mathrm{CZK} \) together. In April there was on store price adjustment. The price of the shorts decreased by \( 10\% \) and the price of the T-shirt increased by \( 10\% \). So the April price of both together the shorts and the T-shirt was by \( 20\,\mathrm{CZK} \) lower. What was the April price of the T-shirt?
\( 220\,\mathrm{CZK} \)
\( 200\,\mathrm{CZK} \)
\( 180\,\mathrm{CZK} \)
\( 400\,\mathrm{CZK} \)