Question

The cost of 3 rackets and 2 squash balls is £26.57. The cost of 5 rackets and 7 squash balls is
The cost of 3 rackets and 2 squash balls is £26.57.
The cost of 5 rackets and 7 squash balls is £52.90.
Work out the cost of
a) a squash ball.
b) a racket

Answer 1

Taking into account the definition of a system of linear equations:

• the cost of a squash ball and a racket are £2.35 and £7.29 respectively.
• the cost of 1 kg of radishes and 1 kg of carrots are £3.1 and £1.6 respectively.

System of linear equations

A system of linear equations is a set of two or more equations of the first degree, in which two or more unknowns are related.

Solving a system of equations consists of finding the value of each unknown so that all the equations of the system are satisfied. That is to say, the values ​​of the unknowns must be sought, with which when replacing, they must give the solution proposed in both equations.

Cost of a squash ball and a racket

In this case, a system of linear equations must be proposed taking into account that:

• "x" represents the cost of a squash ball.
• "y" represents the cost of a racket.

The cost of 3 rackets and 2 squash balls is £26.57.

The cost of 5 rackets and 7 squash balls is £52.90.

So, the system of equations to be solved is

$\left \{ {{2x+3y=26.57} \atop {7x+5y=52.90}} \right.$

There are several methods to solve a system of equations, it is decided to solve it using the substitution method, which consists of clearing one of the two variables in one of the equations of the system and substituting its value in the other equation.

In this case, isolating the variable x from the first equation:

$x=\frac{26.57 - 3y}{2}$

Substituting the expression in the second you get:

$7\frac{26.57 - 3y}{2}+5y=52.90$

Solving:

$\frac{7}{2}(26.57 - 3y)+5y=52.90$

$\frac{7}{2}x26.57 - \frac{7}{2}x3y+5y=52.90$

92.995 - 10.5y +5y= 52.90

92.995 - 5.5y= 52.90

- 5.5y= 52.90 -92.995

- 5.5y= -40.095

y= (-40.095)÷ (-5.5)

y= 7.29

So: $x=\frac{26.57 - 3y}{2}=\frac{26.57 - 3x7.29}{2}$

x= 2.35

Finally, the cost of a squash ball and a racket are £2.35 and £7.29 respectively.

Cost of 1 kg of radishes and 1 kg of carrots

In this case, a system of linear equations must be proposed taking into account that:

• "x" represents the cost of 1 kg of radishes.
• "y" represents the cost of 1 kg of carrots.

The cost of 4 kg of radishes and 1.5 kg of carrots is £14.80.

The cost of 3 kg of radishes and 2 kg of carrots is £12.50.

So, the system of equations to be solved is

$\left \{ {{4x+1.5y=14.80} \atop {3x+2y=12.50}} \right.$

You can solve this system using the substitution method. Isolating the variable x from the first equation:

$x=\frac{14.80 - 1.5y}{4}$

Substituting the expression in the second you get:

$3\frac{14.80 - 1.5y}{4}+2y=12.50$

Solving:

$\frac{3}{4}(14.80 - 1.5y)+2y=12.50$

$\frac{3}{4}x14.80 - \frac{3}{4}x1.5y+2y=12.50$

11.1 - 1.125y +2y= 12.50

- 1.125y +2y= 12.50 - 11.1

0.875y=  1.4

y= 1.4 ÷0.875

y=1.6

So: $x=\frac{14.80 - 1.5y}{4}=\frac{14.80 - 1.5x1.6}{4}$

x= 3.1

Finally, the cost of 1 kg of radishes and 1 kg of carrots are £3.1 and £1.6 respectively.

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