Using a pair of equations is useful way to solve a number of mathematical problems. If something can be computed in two different ways, it is a clear hint that the problem can be solved with a pair of equations. In applications of equation pairs, one variable is often denoted by [[$ x $]] and the other by [[$ y $]].

Example 1

The catering service charges [[$ 5,50 $]]​ QAR for a children's portion and[[$ 13,20 $]] QAR for an adults' portion​. At a party, [[$ 130 $]]​ portions are served. The total cost of these portions is [[$ 1500,40 $]]​ QAR. How many people got a children's portion? How many people got an adults' portion?

Solution:

Let the number of children's portions be [[$ x $]] and the number of adults' portions be [[$ y $]]. Based on the data, a pair of equations can be formed.

[[$ \begin{equation} \begin{cases} x + y = 130 \;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;{\color{green} {\text {There are a total of 130 portions.}}}  \\ 5,50x + 13,20y = 1500,40 \;\;\;\;{\color{green} {\text {The children's portion costs 5,50 QAR and the adults' portion 13,20 QAR.}}}\\ \end{cases} \end{equation}  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\color{green} {\text {The total price of all servings is 1500,40 QAR.}}} $]]​

Solve the pair of equations using the substitution method.

[[$ \begin{align*} 5,50x + 13,20 {\color{red} {\: \cdot \: (130 - x)} } &= 1500,40 \\ 5,50x + 1716 - 13,20x &= 1500,40 \\ 5,50x - 13,20x &= 1500,40 - 1716 \\ -7,7x &= -215,60 \;\;\;\;\;\;\;\;\;\;\; ||:\:(-7,7) \\ x &= 28 \end{align*} $]]​

Place the obtained solution of [[$ x $]] in the equation in order to solve [[$ y $]].

[[$ y = 130 - x = 130 - 28 = 102 $]]​

Answer: There were [[$ 28 $]] children's portions and [[$ 102 $]] adults' portions.

Example 2

Juice concentrate and water must be mixed in a ratio of [[$ 1: 9 $]]. [[$ 30 $]] liters of juice are needed for a birthday party. How much juice concentrate and water is needed?

Solution:

Denote the amount of juice concentrate by [[$ x $]] and the amount of water by [[$ y $]]. Form a pair of equations.

[[$ \begin{equation} \begin{cases}  x + y = 30 \;\;\;\; {\color{green} {\text {There are a total of 30 liters of mixed drink.}}}  \\ \displaystyle\frac {x} {y} = \displaystyle\frac {1} {9} \;\;\;\;\;\;\;\;\; {\color{green} {\text {Mixing ratio of juice and water.}}}\\ \end{cases} \end{equation} $]]​

Convert the equation that describes the mixing ratio to another form. Place the resulting value of [[$ y $]] in the upper equation.

[[$ \begin{align*} x + {\color{red} {9x} } &= 30 \\ 10x &= 30 \;\;\;\; ||:\:10 \\ x &= 3\end{align*} $]]​

Place the obtained value for [[$ x $]] in the equation in order to solve the value of [[$ y $]].

[[$ y = 9x = 9 \cdot 3 = 27 $]]​

Answer: [[$ 3 $]] liters of juice concentrate and [[$ 27 $]] liters of water are needed.