Multiply one or both of the equations through to create ± the same coefficient of one of the variables. In this example, it is easier to match the \(y\) terms as it only requires one of the equations to be multiplied.
In this example, we can't avoid multiplying both equations. We could do 5\(\times\)① and 3\(\times\)② to eliminate \(x\) or 3\(\times\)① and 4\(\times\)② to eliminate \(y\). It doesn't really matter which we choose. Let's eliminate \(y\):
Now, because the coefficients of \(y\) have the same signs, we subtract the two equations to eliminate \(y\). Subtracting ③ – ④ would give two negatives, so it might be tidier to subtract upside down and do ④ – ③ instead:
$$
\begin{eqnarray}
④-③:\:\:11x &=& 33\\[6pt]
x &=& 3
\end{eqnarray}
$$
Now we choose the easier looking of the original equations and substitute \(x=3\) into it to find \(y\):
Two families buy cinema tickets. Mr Taylor buys 2 adult and 2 child tickets for £23.50. Ms Clyde buys 1 adult and 3 child tickets for £20.25. Write down two equations to represent this information and solve them to find the price of one adult ticket and the price of one child ticket.
Using \(a\) for the price in £ of an adult ticket and \(c\) for the price in £ of a child ticket, we have:
So an adult ticket costs £7.50 and a child ticket costs £4.25.
Example 9 (non-calculator)
SQA National 5 Maths 2016 P1 Q4
Charlie is making costumes for a school show. One day he made 2 cloaks and 3 dresses. The total amount of material he used was 9.6 square metres. (a) Write down an equation to illustrate this information. (b) The following day Charlie made 3 cloaks and 4 dresses. The total amount of material he used was 13.3 square metres. Write down an equation to illustrate this information. (c) Calculate the amount of material required to make one cloak and the amount of material required to make one dress.
(a) We need to choose our letters. Let's use \(c\) for the amount of material in a cloak and \(d\) for the amount of material in a dress.
So we get the equation: \(2c+3d=9.6\)
(b) The second equation is \(3c+4d=13.3\)
(c) Now we need to solve the two equations simultaneously.
So a cloak uses 1.5 square metres of material and a dress uses 2.2 square metres of material.
Example 10 (non-calculator)
SQA National 5 Maths 2019 P1 Q8
John bought 7 bags of cement and 3 bags of gravel. The total weight of these bags was 215 kilograms. (a) Write down an equation to illustrate this information.
Shona bought 5 bags of cement and 4 bags of gravel. The total weight of her bags was 200 kilograms. (b) Write down an equation to illustrate this information. (c) Calculate the weight of one bag of cement and the weight of one bag of gravel.
(a) We need to choose our letters. Let's use \(c\) for the weight of one bag of cement and \(g\) for the weight of one bag of gravel.
So we get the equation: \(7c+3g=215\)
(b) The second equation is \(5c+4g=200\)
(c) Now we need to solve the two equations simultaneously.
$$
\begin{align}
③-④:\:\:\:\:\:\:13c &= 260\\[6pt]
c &= \small\frac{260}{13}\\[6pt]
c &= 20\\[18pt]
\small\textsf{Subs }\normalsize①:\:\:\:\:7(20)+3g &= 215\\[6pt]
140+3g &= 215\\[6pt]
3g &= 215-140\\[6pt]
3g &= 75\\[6pt]
g &= \small\frac{75}{3}\\[6pt]
g &= 25\\[6pt]
\end{align}
$$
So one bag of cement weighs 20 kg and one bag of gravel weighs 25 kg.
Example 11 (calculator)
SQA National 5 Maths 2022 P2 Q4
Moira buys 4 mangoes and 3 apples at a fruit shop. The total cost is £4.25. (a) Write down an equation to illustrate this information.
Sami buys 5 mangoes and 2 apples in the same fruit shop. The total cost is £4.70. (b) Write down an equation to illustrate this information. (c) Calculate, algebraically, the cost of a mango and the cost of an apple.
(a) We need to choose our letters. Let's use \(m\) for the price of one mango and \(a\) for the price of one apple. We can work in pounds or pence in this question, but to avoid decimals, perhaps it's easier to work in pence.
So we get the equation: \(4m+3a=425\)
(b) The second equation is \(5m+2a=470\)
(c) Now we need to solve the two equations simultaneously.
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