Construction of Simultaneous Equations (SQA National 5 Maths): Revision Note

Always check that you've answered the question!

Sometimes finding x and y isn't the end. E.g. you may have to state a conclusion.

You don't need to use x and y as your variables.

If an exam question asks you to use a particular set of letters, be sure to follow those instructions.

At a bakery, Eilidh pays £9 in total for six bagels and twelve chocolate bites.

Let $b$ be the price of a bagel and $c$ be the price of a chocolate bite.

(a) Write down an equation in $b$ and $c$ to illustrate this information.
 

At the same bakery, Iain buys nine bagels and ten chocolate bites.

His total cost is £12.30.

(b) Write down an equation in $b$ and $c$ to illustrate this information.
 

Bevan buys 5 bagels and 15 chocolate bites from the bakery.

(c) Calculate the total cost of Bevan's purchase.
 

Answer:

Part (a)

The two variables are the price of a bagel, $b$, and the price of a chocolate bite, $c$

Write an equation for the Eilidh's purchase, using prices in pounds

• You could use prices in pence instead

• In that case you would have to convert £9 to 900p for the right hand side of the equation

$6b+12c=9$
 

Part (b)

Write an equation for Iain's purchase, using prices in pounds

• You could use prices in pence instead

• In that case you would have to convert £12.30 to 1230p for the right hand side of the equation

• Make sure you use the same units for both equations!

$9b+10c=12.3$
 

Part (c)

Label the equations 1 and 2

$\overline{)1}6b+12c=9$

$\overline{)2}9b+10c=12.3$

'Rescale' the equations to eliminate the $b$ terms

• You could also choose to eliminate the $c$ terms first

• In that case you would multiply equation 1 by 5 and equation 2 by 6 before subtracting the equations

Make the $b$ terms equal by

• multiplying all parts of equation 1 by 3

• and all parts of equation 2 by 2

Label these as equations 3 and 4

$\begin{array}{rcl}\overline{)1}\times 318b+36c& =& 27\overline{)3}\\ \overline{)2}\times 218b+20c& =& 24.6\overline{)4}\end{array}$

To eliminate $b$, subtract equation 4 from equation 3

$\overline{)3}-\overline{)4}$ $$\begin{gathered} 18b+36c=27 \\ \underline{-\left(18b+20c=24.6\right)} \\ 16c=2.4 \end{gathered}$$

Solve for $c$

$c=\frac{2.4}{16}=0.15$

Substitute this into either equation to find $b$

• Here we will use equation 1

$\begin{array}{rcl}\overline{)1}6b+12\left(0.15\right)& =& 9\\ 6b+1.8& =& 9\\ 6b& =& 7.2\\ b& =& 1.2\end{array}$

So chocolate bites cost £0.15 each and bagels cost £1.20 each

Use these values to find the price of 5 bagels and 15 chocolate bites

$\left(5\times 1.2\right)+\left(15\times 0.15\right)=8.25$

£8.25